{"id":3102,"date":"2018-12-11T13:51:03","date_gmt":"2018-12-11T18:51:03","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/factor-special-products\/"},"modified":"2018-12-11T13:51:03","modified_gmt":"2018-12-11T18:51:03","slug":"factor-special-products","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/factor-special-products\/","title":{"raw":"Factor Special Products","rendered":"Factor Special Products"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Factor perfect square trinomials<\/li><li>Factor differences of squares<\/li><li>Factor sums and differences of cubes<\/li><\/ul><\/div><div data-type=\"note\" class=\"be-prepared\"><p>Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167835280232\" type=\"1\"><li>Simplify: \\({\\left(3{x}^{2}\\right)}^{3}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb#fs-id1167835304261\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Multiply: \\({\\left(m+4\\right)}^{2}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Multiply: \\(\\left(x-3\\right)\\left(x+3\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836717042\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><p id=\"fs-id1167835423012\">We have seen that some binomials and trinomials result from special products\u2014squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.<\/p><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167830769807\"><h3 data-type=\"title\">Factor Perfect Square Trinomials<\/h3><p>Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter.<\/p><span data-type=\"media\" data-alt=\"In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.\"><\/span><p>The trinomial \\(9{x}^{2}+24x+16\\) is called a <em data-effect=\"italics\">perfect square trinomial<\/em>. It is the square of the binomial \\(3x+4.\\)<\/p><p id=\"fs-id1167835367325\">In this chapter, you will start with a perfect square trinomial and factor it into its <span data-type=\"term\" class=\"no-emphasis\">prime<\/span> factors.<\/p><p>You could factor this <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span> using the methods described in the last section, since it is of the form \\(a{x}^{2}+bx+c.\\) But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.<\/p><p id=\"fs-id1167834120157\">Here is the pattern\u2014the reverse of the binomial squares pattern.<\/p><div data-type=\"note\" id=\"fs-id1167832076542\"><div data-type=\"title\">Perfect Square Trinomials Pattern<\/div><p id=\"fs-id1167832153960\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers<\/p><div data-type=\"equation\" id=\"fs-id1167834535644\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill {a}^{2}+2ab+{b}^{2}={\\left(a+b\\right)}^{2}\\hfill \\\\ \\hfill {a}^{2}-2ab+{b}^{2}={\\left(a-b\\right)}^{2}\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167834430982\">To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, \\({a}^{2}.\\) Next check that the last term is a perfect square, \\({b}^{2}.\\) Then check the middle term\u2014is it the product, \\(2ab?\\) If everything checks, you can easily write the factors.<\/p><div data-type=\"example\" id=\"fs-id1167832056529\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Factor Perfect Square Trinomials<\/div><div data-type=\"exercise\" id=\"fs-id1167834533424\"><div data-type=\"problem\" id=\"fs-id1167831911097\"><p id=\"fs-id1167835352469\">Factor: \\(9{x}^{2}+12x+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306597\"><span data-type=\"media\" id=\"fs-id1167835322243\" data-alt=\"Step 1 is to check if the trinomial fits the perfect square trinomials pattern, a squared plus 2ab plus b squared. For this we check if the first term is a perfect square. 9 x squared is the square of 3x. Next we check if the last term is a perfect square. 4 is the square of 2. Next we check if the middle term is 2ab. 12 x is twice 3x times 2. Hence we have a perfect square trinomial.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to check if the trinomial fits the perfect square trinomials pattern, a squared plus 2ab plus b squared. For this we check if the first term is a perfect square. 9 x squared is the square of 3x. Next we check if the last term is a perfect square. 4 is the square of 2. Next we check if the middle term is 2ab. 12 x is twice 3x times 2. Hence we have a perfect square trinomial.\"><\/span><span data-type=\"media\" id=\"fs-id1167835335949\" data-alt=\"Step 2 is to write this as the square of a binomial. We write it as open parentheses 3x plus 2 close parentheses squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to write this as the square of a binomial. We write it as open parentheses 3x plus 2 close parentheses squared.\"><\/span><span data-type=\"media\" data-alt=\"Step 3 is to check by multiplying.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to check by multiplying.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167827940329\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835364113\"><p id=\"fs-id1167835263503\">Factor: \\(4{x}^{2}+12x+9.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832082000\"><p id=\"fs-id1167835238962\">\\({\\left(2x+3\\right)}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835369810\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831883553\"><div data-type=\"problem\" id=\"fs-id1167828421283\"><p id=\"fs-id1167834189852\">Factor: \\(9{y}^{2}+24y+16.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826857212\"><p id=\"fs-id1167831191480\">\\({\\left(3y+4\\right)}^{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835417985\">The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern \\({a}^{2}-2ab+{b}^{2},\\) which factors to \\({\\left(a-b\\right)}^{2}.\\)<\/p><p id=\"fs-id1167832068338\">The steps are summarized here.<\/p><div data-type=\"note\" id=\"fs-id1167834505010\" class=\"howto\"><div data-type=\"title\">Factor perfect square trinomials.<\/div><p id=\"fs-id1167834213922\">\\(\\begin{array}{cccccccc}\\mathbf{\\text{Step 1.}}\\hfill &amp; \\text{Does the trinomial fit the pattern?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{a}^{2}+2ab+{b}^{2}\\hfill &amp; &amp; &amp; \\hfill {a}^{2}-2ab+{b}^{2}\\hfill \\\\ &amp; \\text{Is the first term a perfect square?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a\\right)}^{2}\\hfill \\\\ &amp; \\text{Write it as a square.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\\\ &amp; \\text{Is the last term a perfect square?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}\\phantom{\\rule{4.5em}{0ex}}{\\left(b\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a\\right)}^{2}\\phantom{\\rule{4.5em}{0ex}}{\\left(b\\right)}^{2}\\hfill \\\\ &amp; \\text{Write it as a square.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\\\ &amp; \\text{Check the middle term. Is it}\\phantom{\\rule{0.2em}{0ex}}2ab?\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}{}_{\\text{\u2198}}\\underset{2\u00b7a\u00b7b}{}{}_{\\text{\u2199}}{\\left(b\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a\\right)}^{2}{}_{\\text{\u2198}}\\underset{2\u00b7a\u00b7b}{}{}_{\\text{\u2199}}{\\left(b\\right)}^{2}\\hfill \\\\ \\mathbf{\\text{Step 2.}}\\hfill &amp; \\text{Write the square of the binomial.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a+b\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a-b\\right)}^{2}\\hfill \\\\ \\mathbf{\\text{Step 3.}}\\hfill &amp; \\text{Check by multiplying.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><p id=\"fs-id1167835377433\">We\u2019ll work one now where the middle term is negative.<\/p><div data-type=\"example\" id=\"fs-id1167835345249\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834382630\"><div data-type=\"problem\" id=\"fs-id1167831811607\"><p>Factor: \\(81{y}^{2}-72y+16.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834194750\"><p id=\"fs-id1167834229171\">The first and last terms are squares. See if the middle term fits the pattern of a <span data-type=\"term\" class=\"no-emphasis\">perfect square<\/span> trinomial. The middle term is negative, so the binomial square would be \\({\\left(a-b\\right)}^{2}.\\)<\/p><table class=\"unnumbered unstyled\" summary=\"The trinomial is 81 y squared minus 72y plus 16. The first and last terms are perfect squares of 9y and 4. Check the middle term. It is twice 9y times 4. The trinomial matches a minus b the whole squared. So we write it as the square of a binomial open parentheses 9y minus 4 close parentheses squared. Finally, we check by multiplying.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832053914\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Are the first and last terms perfect squares?\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834301208\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check the middle term.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831985779\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Does it match \\({\\left(a-b\\right)}^{2}?\\) Yes.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834130259\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write as the square of a binomial.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228768\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check by multiplying:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\hfill {\\left(9y-4\\right)}^{2}\\hfill \\\\ \\hfill {\\left(9y\\right)}^{2}-2\u00b79y\u00b74+{4}^{2}\\hfill \\\\ \\hfill 81{y}^{2}-72y+16\u2713\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835257249\"><div data-type=\"problem\" id=\"fs-id1167831928925\"><p id=\"fs-id1167835258025\">Factor: \\(64{y}^{2}-80y+25.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834505103\"><p id=\"fs-id1167834189773\">\\({\\left(8y-5\\right)}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835235994\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832152848\"><div data-type=\"problem\"><p id=\"fs-id1167831882541\">Factor: \\(16{z}^{2}-72z+81.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835361589\"><p id=\"fs-id1167832044141\">\\({\\left(4z-9\\right)}^{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835330168\">The next example will be a perfect square trinomial with two variables.<\/p><div data-type=\"example\" id=\"fs-id1167835254902\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835193710\"><div data-type=\"problem\" id=\"fs-id1167834061377\"><p id=\"fs-id1167835274949\">Factor: \\(36{x}^{2}+84xy+49{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832052324\"><table id=\"fs-id1167835353428\" class=\"unnumbered unstyled\" summary=\"The trinomial is 36 x squared plus 84xy plus 49 y squared. We test each term to verify the pattern. The trinomial is open parentheses 6x close parentheses squared plus 2 times 6x times 7y plus open parentheses 7y close parentheses squared. We factor to get open parentheses 6x plus 7y close parentheses squared.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834432988\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Test each term to verify the pattern.\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835338542\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233638\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check by multiplying.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\hfill {\\left(6x+7y\\right)}^{2}\\hfill \\\\ \\hfill {\\left(6x\\right)}^{2}+2\u00b76x\u00b77y+{\\left(7y\\right)}^{2}\\hfill \\\\ \\hfill 36{x}^{2}+84xy+49{y}^{2}\u2713\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834301082\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835231699\">Factor: \\(49{x}^{2}+84xy+36{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835344675\"><p id=\"fs-id1167832152791\">\\({\\left(7x+6y\\right)}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835345375\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832057915\"><div data-type=\"problem\" id=\"fs-id1167831893298\"><p id=\"fs-id1167835375915\">Factor: \\(64{m}^{2}+112mn+49{n}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834372379\"><p id=\"fs-id1167832152694\">\\({\\left(8m+7n\\right)}^{2}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835357616\">Remember the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a <span data-type=\"term\" class=\"no-emphasis\">GCF<\/span> in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.<\/p><div data-type=\"example\" id=\"fs-id1167834396304\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835370069\"><div data-type=\"problem\" id=\"fs-id1167831882219\"><p id=\"fs-id1167834065318\">Factor: \\(100{x}^{2}y-80xy+16y.\\)<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167835513206\" class=\"unnumbered unstyled\" summary=\"Is there a GCF in 100 x squared y minus 80xy plus 16y? Yes. Factoring it out, we get 4y open parentheses 25 x squared minus 20x plus 4 close parentheses. Is this a perfect square trinomial? To verify the pattern, we rewrite as 4y open bracket open parentheses 5x close parentheses squared minus 2 times 5x times 2 plus 2 squared close bracket. Factor to get 4y open parentheses 5x minus 2 close parentheses squared. Finally, we check by multiplying.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834196287\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Is there a GCF? Yes, \\(4y,\\) so factor it out.\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832152888\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Is this a perfect square trinomial?<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Verify the pattern.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834329838\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p>Remember: Keep the factor 4<em data-effect=\"italics\">y<\/em> in the final product.<\/p><p id=\"fs-id1167832086884\">Check:<\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{4em}{0ex}}\\begin{array}{c}\\hfill 4y{\\left(5x-2\\right)}^{2}\\hfill \\\\ \\hfill 4y\\left[{\\left(5x\\right)}^{2}-2\u00b75x\u00b72+{2}^{2}\\right]\\hfill \\\\ \\hfill 4y\\left(25{x}^{2}-20x+4\\right)\\hfill \\\\ \\hfill 100{x}^{2}y-80xy+16y\u2713\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835348534\"><div data-type=\"problem\" id=\"fs-id1167834557137\"><p>Factor: \\(8{x}^{2}y-24xy+18y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834193624\"><p id=\"fs-id1167832075544\">\\(2y{\\left(2x-3\\right)}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167830705949\"><p>Factor: \\(27{p}^{2}q+90pq+75q.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835349497\">\\(3q{\\left(3p+5\\right)}^{2}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Factor Differences of Squares<\/h3><p id=\"fs-id1167834185224\">The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here\u2019s an example:<\/p><span data-type=\"media\" id=\"fs-id1167826987977\" data-alt=\"We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.\"><\/span><p>A difference of squares factors to a product of conjugates.<\/p><div data-type=\"note\" id=\"fs-id1167834063594\"><div data-type=\"title\">Difference of Squares Pattern<\/div><p id=\"fs-id1167834124313\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p><span data-type=\"media\" data-alt=\"a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are conjugates.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are conjugates.\"><\/span><\/div><p id=\"fs-id1167826857435\">Remember, \u201cdifference\u201d refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Factor a Trinomial Using the Difference of Squares<\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835326544\"><p>Factor: \\(64{y}^{2}-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832226786\"><span data-type=\"media\" id=\"fs-id1167835344526\" data-alt=\"Step 1 is to check if the binomial 64 y squared minus 1 fits the pattern. For that we check the following: Is this a difference? Yes. Are the first and last terms perfect squares? Yes.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to check if the binomial 64 y squared minus 1 fits the pattern. For that we check the following: Is this a difference? Yes. Are the first and last terms perfect squares? Yes.\"><\/span><span data-type=\"media\" id=\"fs-id1167835417524\" data-alt=\"Step 2 is to write both terms as squares, So, we have open parentheses 8y close parentheses squared minus 1 squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to write both terms as squares, So, we have open parentheses 8y close parentheses squared minus 1 squared.\"><\/span><span data-type=\"media\" id=\"fs-id1167835380209\" data-alt=\"Step 3 is to write the product of conjugates 8y minus 1, 8y plus 1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to write the product of conjugates 8y minus 1, 8y plus 1.\"><\/span><span data-type=\"media\" id=\"fs-id1167832075154\" data-alt=\"Step 4 is to check. We multiply to get the original binomial\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check. We multiply to get the original binomial\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835596424\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834130741\"><div data-type=\"problem\" id=\"fs-id1167835303608\"><p id=\"fs-id1167834085029\">Factor: \\(121{m}^{2}-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831112564\"><p id=\"fs-id1167834463142\">\\(\\left(11m-1\\right)\\left(11m+1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834429753\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834522827\"><div data-type=\"problem\" id=\"fs-id1167835324763\"><p id=\"fs-id1167834324728\">Factor: \\(81{y}^{2}-1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835420022\"><p id=\"fs-id1167835299916\">\\(\\left(9y-1\\right)\\left(9y+1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835369387\" class=\"howto\"><div data-type=\"title\">Factor differences of squares.<\/div><p id=\"fs-id1167831922306\">\\(\\begin{array}{ccccc}\\mathbf{\\text{Step 1.}}\\hfill &amp; \\text{Does the binomial fit the pattern?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{a}^{2}-{b}^{2}\\hfill \\\\ &amp; \\text{Is this a difference?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\text{____}-\\text{____}\\hfill \\\\ &amp; \\text{Are the first and last terms perfect squares?}\\hfill &amp; &amp; &amp; \\\\ \\mathbf{\\text{Step 2.}}\\hfill &amp; \\text{Write them as squares.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}-{\\left(b\\right)}^{2}\\hfill \\\\ \\mathbf{\\text{Step 3.}}\\hfill &amp; \\text{Write the product of conjugates.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(a-b\\right)\\left(a+b\\right)\\hfill \\\\ \\mathbf{\\text{Step 4.}}\\hfill &amp; \\text{Check by multiplying.}\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><p id=\"fs-id1167831116937\">It is important to remember that <em data-effect=\"italics\">sums of squares do not factor into a product of binomials<\/em>. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression \\({a}^{2}+{b}^{2}\\) is prime!<\/p><p id=\"fs-id1171791716465\">The next example shows variables in both terms.<\/p><div data-type=\"example\" id=\"fs-id1167834228380\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835257523\"><div data-type=\"problem\" id=\"fs-id1167834228385\"><p id=\"fs-id1167832042469\">Factor: \\(144{x}^{2}-49{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831944012\"><p>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}144{x}^{2}-49{y}^{2}\\hfill \\\\ \\text{Is this a difference of squares? Yes.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}{\\left(12x\\right)}^{2}-{\\left(7y\\right)}^{2}\\hfill \\\\ \\text{Factor as the product of conjugates.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{4em}{0ex}}\\left(12x-7y\\right)\\left(12x+7y\\right)\\hfill \\\\ \\text{Check by multiplying.}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}\\left(12x-7y\\right)\\left(12x+7y\\right)\\hfill &amp; &amp; &amp; \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}144{x}^{2}-49{y}^{2}\u2713\\hfill &amp; &amp; &amp; \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834397185\"><div data-type=\"problem\" id=\"fs-id1167832058472\"><p id=\"fs-id1167835371022\">Factor: \\(196{m}^{2}-25{n}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834162074\"><p id=\"fs-id1167834526533\">\\(\\left(16m-5n\\right)\\left(16m+5n\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835349317\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831103990\"><div data-type=\"problem\"><p id=\"fs-id1167835171108\">Factor: \\(121{p}^{2}-9{q}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835381273\"><p id=\"fs-id1167835308295\">\\(\\left(11p-3q\\right)\\left(11p+3q\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167834300692\">As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may \u201cdisguise\u201d the difference of squares and you won\u2019t recognize the perfect squares until you factor the GCF.<\/p><p id=\"fs-id1167826819804\">Also, to completely factor the binomial in the next example, we\u2019ll factor a difference of squares twice!<\/p><div data-type=\"example\" id=\"fs-id1167832042561\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835514442\"><div data-type=\"problem\" id=\"fs-id1167826880317\"><p id=\"fs-id1167835317851\">Factor: \\(48{x}^{4}{y}^{2}-243{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831871375\"><p id=\"fs-id1167834064508\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\hfill 48{x}^{4}{y}^{2}-243{y}^{2}\\hfill \\\\ \\text{Is there a GCF? Yes,}\\phantom{\\rule{0.2em}{0ex}}3{y}^{2}\\text{\u2014factor it out!}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 3{y}^{2}\\left(16{x}^{4}-81\\right)\\hfill \\\\ \\text{Is the binomial a difference of squares? Yes.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 3{y}^{2}\\left({\\left(4{x}^{2}\\right)}^{2}-{\\left(9\\right)}^{2}\\right)\\hfill \\\\ \\text{Factor as a product of conjugates.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 3{y}^{2}\\left(4{x}^{2}-9\\right)\\left(4{x}^{2}+9\\right)\\hfill \\\\ \\text{Notice the first binomial is also a difference of squares!}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 3{y}^{2}\\left({\\left(2x\\right)}^{2}-{\\left(3\\right)}^{2}\\right)\\left(4{x}^{2}+9\\right)\\hfill \\\\ \\text{Factor it as the product of conjugates.}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill 3{y}^{2}\\left(2x-3\\right)\\left(2x+3\\right)\\left(4{x}^{2}+9\\right)\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167832226903\">The last factor, the sum of squares, cannot be factored.<\/p><p id=\"fs-id1167832015899\">\\(\\begin{array}{c}\\text{Check by multiplying:}\\hfill \\\\ \\\\ \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}3{y}^{2}\\left(2x-3\\right)\\left(2x+3\\right)\\left(4{x}^{2}+9\\right)\\hfill \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}3{y}^{2}\\left(4{x}^{2}-9\\right)\\left(4{x}^{2}+9\\right)\\hfill \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}3{y}^{2}\\left(16{x}^{4}-81\\right)\\hfill \\\\ \\hfill \\phantom{\\rule{4em}{0ex}}48{x}^{4}{y}^{2}-243{y}^{2}\u2713\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834423133\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834423136\"><div data-type=\"problem\" id=\"fs-id1167835361669\"><p id=\"fs-id1167835361671\">Factor: \\(2{x}^{4}{y}^{2}-32{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830700962\"><p id=\"fs-id1167830700964\">\\(2{y}^{2}\\left(x-2\\right)\\left(x+2\\right)\\left({x}^{2}+4\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835374550\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835174151\"><div data-type=\"problem\" id=\"fs-id1167835420262\"><p id=\"fs-id1167835420264\">Factor: \\(7{a}^{4}{c}^{2}-7{b}^{4}{c}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830960797\"><p id=\"fs-id1167830960799\">\\(7{c}^{2}\\left(a-b\\right)\\left(a+b\\right)\\left({a}^{2}+{b}^{2}\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835288115\">The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.<\/p><div data-type=\"example\" id=\"fs-id1167835377553\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831894420\"><div data-type=\"problem\" id=\"fs-id1167831894422\"><p id=\"fs-id1167834532547\">Factor: \\({x}^{2}-6x+9-{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831910960\"><p id=\"fs-id1167826778526\">Notice that the first three terms form a perfect square trinomial.<\/p><table id=\"fs-id1167832076022\" class=\"unnumbered unstyled\" summary=\"We have x squared minus 6x plus 9 minus y squared. We factor by grouping the first three terms. Use the perfect square trinomial pattern to get open parentheses x minus 3 close parentheses squared minus y squared. Is this a difference of squares? Yes. Factor as the product of conjugates, open parentheses x minus 3 minus y close parentheses open parentheses x minus 3 plus y close parentheses. You may want to rewrite the solution as open parentheses x minus y minus 3 close parentheses open parentheses x plus y minus 3 close parentheses.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835310550\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor by grouping the first three terms.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835339144\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the perfect square trinomial pattern.\u2003\u2003\u2003\u2003\u2003<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832211979\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Is this a difference of squares? Yes.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Yes\u2014write them as squares.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826802852\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor as the product of conjugates.<\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830865387\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834473752\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834299522\">You may want to rewrite the solution as \\(\\left(x-y-3\\right)\\left(x+y-3\\right).\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834213904\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832041469\"><div data-type=\"problem\" id=\"fs-id1167832041472\"><p id=\"fs-id1167832041474\">Factor: \\({x}^{2}-10x+25-{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835329671\"><p id=\"fs-id1167835329674\">\\(\\left(x-5-y\\right)\\left(x-5+y\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834340149\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167828434986\"><div data-type=\"problem\" id=\"fs-id1167828434988\"><p id=\"fs-id1167835318775\">Factor: \\({x}^{2}+6x+9-4{y}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832036135\"><p id=\"fs-id1167832036137\">\\(\\left(x+3-2y\\right)\\left(x+3+2y\\right)\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834137540\"><h3 data-type=\"title\">Factor Sums and Differences of Cubes<\/h3><p id=\"fs-id1167834137546\">There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.<\/p><div data-type=\"equation\" id=\"fs-id1167835410207\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill {a}^{3}+{b}^{3}=\\left(a+b\\right)\\left({a}^{2}-ab+{b}^{2}\\right)\\hfill \\\\ \\hfill {a}^{3}-{b}^{3}=\\left(a-b\\right)\\left({a}^{2}+ab+{b}^{2}\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167834222369\">We\u2019ll check the first pattern and leave the second to you.<\/p><table id=\"fs-id1167835303688\" class=\"unnumbered unstyled\" summary=\"Open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared. Distribute: a open parentheses a squared minus ab plus b squared plus b open parentheses a squared minus ab plus b squared close parentheses. Multiply: a cubed minus a squared b plus ab squared plus a squared b minus ab squared plus b cubed. Combine like terms: a cubed plus b cubed.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826937760\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Distribute.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834517541\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835421924\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><div data-type=\"note\" id=\"fs-id1167835353807\"><div data-type=\"title\">Sum and Difference of Cubes Pattern<\/div><div data-type=\"equation\" id=\"fs-id1167830865652\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill {a}^{3}+{b}^{3}=\\left(a+b\\right)\\left({a}^{2}-ab+{b}^{2}\\right)\\hfill \\\\ \\hfill {a}^{3}-{b}^{3}=\\left(a-b\\right)\\left({a}^{2}+ab+{b}^{2}\\right)\\hfill \\end{array}\\)<\/div><\/div><p id=\"fs-id1167835483694\">The two patterns look very similar, don\u2019t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.<\/p><span data-type=\"media\" id=\"fs-id1167835483696\" data-alt=\"a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.\"><\/span><p id=\"fs-id1167834133012\">The trinomial factor in the sum and difference of cubes pattern cannot be factored.<\/p><p id=\"fs-id1167834133016\">It be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in <a href=\"#fs-id1167835337696\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><table id=\"fs-id1167835337696\" summary=\"This table has 11 columns and 2 columns. The first column labels each row n and n cubed. The remaining columns of the first row have the numbers 1 through 10. The remaining columns of the second row have the numbers 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.\"><thead><tr valign=\"top\"><th data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">n<\/em><\/th><th data-valign=\"top\" data-align=\"left\">1<\/th><th data-valign=\"top\" data-align=\"left\">2<\/th><th data-valign=\"top\" data-align=\"left\">3<\/th><th data-valign=\"top\" data-align=\"left\">4<\/th><th data-valign=\"top\" data-align=\"left\">5<\/th><th data-valign=\"top\" data-align=\"left\">6<\/th><th data-valign=\"top\" data-align=\"left\">7<\/th><th data-valign=\"top\" data-align=\"left\">8<\/th><th data-valign=\"top\" data-align=\"left\">9<\/th><th data-valign=\"top\" data-align=\"left\">10<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\({n}^{3}\\)<\/td><td data-valign=\"top\" data-align=\"left\">1<\/td><td data-valign=\"top\" data-align=\"left\">8<\/td><td data-valign=\"top\" data-align=\"left\">27<\/td><td data-valign=\"top\" data-align=\"left\">64<\/td><td data-valign=\"top\" data-align=\"left\">125<\/td><td data-valign=\"top\" data-align=\"left\">216<\/td><td data-valign=\"top\" data-align=\"left\">343<\/td><td data-valign=\"top\" data-align=\"left\">512<\/td><td data-valign=\"top\" data-align=\"left\">729<\/td><td data-valign=\"top\" data-align=\"left\">1000<\/td><\/tr><\/tbody><\/table><div data-type=\"example\" id=\"fs-id1167830963362\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Factor the Sum or Difference of Cubes<\/div><div data-type=\"exercise\" id=\"fs-id1167830963365\"><div data-type=\"problem\" id=\"fs-id1167830963367\"><p id=\"fs-id1167830963369\">Factor: \\({x}^{3}+64.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834516173\"><span data-type=\"media\" id=\"fs-id1167834516175\" data-alt=\"Step 1 is to check if the binomial fits the sum or difference of cubes pattern. For this, we check whether it is a sum or difference. x cubed plus 64 is a sum. Next we check if the first and last terms are perfect cubes. They are\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to check if the binomial fits the sum or difference of cubes pattern. For this, we check whether it is a sum or difference. x cubed plus 64 is a sum. Next we check if the first and last terms are perfect cubes. They are\"><\/span><span data-type=\"media\" id=\"fs-id1167826986850\" data-alt=\"Step 2 is to rewrite as cubes. So we rewrite as x cubed plus 4 cubed.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rewrite as cubes. So we rewrite as x cubed plus 4 cubed.\"><\/span><span data-type=\"media\" id=\"fs-id1167831881792\" data-alt=\"Step 3 is to use either the sum or difference of cubes pattern. Since this is a sum of cubes, we get open parentheses x plus 4 close parentheses open parentheses x squared minus 4x plus 4 squared.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use either the sum or difference of cubes pattern. Since this is a sum of cubes, we get open parentheses x plus 4 close parentheses open parentheses x squared minus 4x plus 4 squared.\"><\/span><span data-type=\"media\" id=\"fs-id1167834367159\" data-alt=\"Step 4 is to simplify inside the parentheses. It is already simplified\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to simplify inside the parentheses. It is already simplified\"><\/span><span data-type=\"media\" id=\"fs-id1167834192367\" data-alt=\"Step 5 is to check by multiplying the factors.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check by multiplying the factors.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826801876\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826801879\"><div data-type=\"problem\" id=\"fs-id1167826801881\"><p id=\"fs-id1167835336557\">Factor: \\({x}^{3}+27.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835299809\"><p id=\"fs-id1167835299811\">\\(\\left(x+3\\right)\\left({x}^{2}-3x+9\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835368092\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835368095\"><div data-type=\"problem\" id=\"fs-id1167831115433\"><p id=\"fs-id1167831115435\">Factor: \\({y}^{3}+8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835370061\"><p id=\"fs-id1167835370063\">\\(\\left(y+2\\right)\\left({y}^{2}-2y+4\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830700252\" class=\"howto\"><div data-type=\"title\">Factor the sum or difference of cubes.<\/div><ol id=\"fs-id1167835282990\" type=\"1\" class=\"stepwise\"><li>Does the binomial fit the sum or difference of cubes pattern?<div data-type=\"newline\"><br><\/div>Is it a sum or difference?<div data-type=\"newline\"><br><\/div>Are the first and last terms perfect cubes?<\/li><li>Write them as cubes.<\/li><li>Use either the sum or difference of cubes pattern.<\/li><li>Simplify inside the parentheses.<\/li><li>Check by multiplying the factors.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167835510228\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835510230\"><div data-type=\"problem\" id=\"fs-id1167831970070\"><p id=\"fs-id1167831970072\">Factor: \\(27{u}^{3}-125{v}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835281405\"><table id=\"fs-id1167835281408\" class=\"unnumbered unstyled\" summary=\"The binomial is 27 u cubed minus 125 v cubed. It is a difference. The first and last terms are perfect cubes. We rewrite as open parentheses 3u close parentheses cubed minus open parentheses 5v close parentheses cubed. Using the difference of cubes pattern, we get open parentheses 3u minus 5v close parentheses open parentheses open parentheses3u close parentheses squared plus 3u times 5v plus open parentheses 5v close parentheses squared close parentheses. We simplify to get open parentheses 3u minus 5v close parentheses open parentheses 9 u squared plus 15uv plus 25 v squared close parentheses. Finally, check by multiplying.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830706026\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">This binomial is a difference. The first and last<div data-type=\"newline\"><br><\/div>terms are perfect cubes.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the terms as cubes.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834525139\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the difference of cubes pattern.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835519133\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834396869\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check by multiplying.<\/td><td data-valign=\"top\" data-align=\"left\">We\u2019ll leave the check to you.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834345963\"><div data-type=\"problem\" id=\"fs-id1167834345965\"><p id=\"fs-id1167834345967\">Factor: \\(8{x}^{3}-27{y}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834131074\"><p id=\"fs-id1167834534784\">\\(\\left(2x-3y\\right)\\left(4{x}^{2}-6xy+9{y}^{2}\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831894757\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831040260\"><div data-type=\"problem\" id=\"fs-id1167831040262\"><p id=\"fs-id1167831040264\">Factor: \\(1000{m}^{3}-125{n}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832053341\"><p id=\"fs-id1167832053343\">\\(\\left(10m-5n\\right)\\left(100{m}^{2}-50mn+25{n}^{2}\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167834534369\">In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.<\/p><div data-type=\"example\" id=\"fs-id1167834534372\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834346225\"><div data-type=\"problem\" id=\"fs-id1167834346227\"><p id=\"fs-id1167834346230\">Factor: \\(6{x}^{3}y+48{y}^{4}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826802403\"><table id=\"fs-id1167826802406\" class=\"unnumbered unstyled\" summary=\"We have 6 x cubed y plus 48 y to the power 4. Factoring the common factor, we rewrite as 6y open parentheses x cubed plus 8 y cubed close parentheses. This binomial is a sum and the first and last terms are perfect cubes. We rewrite as 6y open brackets x cubed plus open parentheses 2y close parentheses cubed close bracket. Using the sum of cubes pattern and simplifying, we get 6y open parentheses x plus 2y close parentheses open parentheses x squared minus 2xy plus 4 y squared close parentheses.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835319053\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Factor the common factor.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831921402\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">This binomial is a sum The first and last<div data-type=\"newline\"><br><\/div>terms are perfect cubes.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the terms as cubes.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835623236\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the sum of cubes pattern.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826987383\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826937937\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167832055056\">Check:<\/p><p id=\"fs-id1167832055059\">To check, you may find it easier to multiply the sum of cubes factors first, then multiply that product by \\(6y.\\) We\u2019ll leave the multiplication for you.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832010232\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832010236\"><div data-type=\"problem\" id=\"fs-id1167832010238\"><p id=\"fs-id1167832010240\">Factor: \\(500{p}^{3}+4{q}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834472825\"><p id=\"fs-id1167834472827\">\\(4\\left(5p+q\\right)\\left(25{p}^{2}-5pq+{q}^{2}\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167828436488\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167828436492\"><div data-type=\"problem\" id=\"fs-id1167828436494\"><p id=\"fs-id1167831836142\">Factor: \\(432{c}^{3}+686{d}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834524246\"><p id=\"fs-id1167834524248\">\\(2\\left(6c+7d\\right)\\left(36{c}^{2}-42cd+49{d}^{2}\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1171791690075\">The first term in the next example is a binomial cubed.<\/p><div data-type=\"example\" id=\"fs-id1167835417027\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835417029\"><div data-type=\"problem\" id=\"fs-id1167826807776\"><p id=\"fs-id1167826807778\">Factor: \\({\\left(x+5\\right)}^{3}-64{x}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832055514\"><table id=\"fs-id1167832055518\" class=\"unnumbered unstyled\" summary=\"The binomial open parentheses x plus 5 close parentheses cubed minus 64 x cubed is a difference. The first and last terms are perfect cubes. Using the difference of cubes pattern and simplifying, we get open parentheses minus 3x plus 5 close parentheses open parentheses 21 x squared plus 30x plus 25 close parentheses. Check by multiplying.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834438875\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">This binomial is a difference. The first and<div data-type=\"newline\"><br><\/div>last terms are perfect cubes.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the terms as cubes.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835371036\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Use the difference of cubes pattern.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831922972\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834463097\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Check by multiplying.<\/td><td data-valign=\"top\" data-align=\"left\">We\u2019ll leave the check to you.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831825658\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831825663\"><div data-type=\"problem\" id=\"fs-id1167831825665\"><p id=\"fs-id1167831825667\">Factor: \\({\\left(y+1\\right)}^{3}-27{y}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834219510\"><p id=\"fs-id1167834219512\">\\(\\left(-2y+1\\right)\\left(13{y}^{2}+5y+1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167827967115\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167827967119\"><div data-type=\"problem\" id=\"fs-id1167827967121\"><p id=\"fs-id1167831913457\">Factor: \\({\\left(n+3\\right)}^{3}-125{n}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835359626\"><p id=\"fs-id1167835359628\">\\(\\left(-4n+3\\right)\\left(31{n}^{2}+21n+9\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830837043\" class=\"media-2\"><p id=\"fs-id1167830837047\">Access this online resource for additional instruction and practice with factoring special products.<\/p><ul id=\"fs-id1167832153726\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37BinomCubes\">Factoring Binomials-Cubes #2<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835414608\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835414615\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Perfect Square Trinomials Pattern:<\/strong> If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167831148856\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\hfill {a}^{2}+2ab+{b}^{2}={\\left(a+b\\right)}^{2}\\hfill \\\\ \\hfill {a}^{2}-2ab+{b}^{2}={\\left(a-b\\right)}^{2}\\hfill \\end{array}\\)<\/div><\/li><li><strong data-effect=\"bold\">How to factor perfect square trinomials.<\/strong><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccccccc}\\text{Step 1.}\\hfill &amp; \\text{Does the trinomial fit the pattern?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{a}^{2}+2ab+{b}^{2}\\hfill &amp; &amp; &amp; \\hfill {a}^{2}-2ab+{b}^{2}\\hfill \\\\ &amp; \\text{Is the first term a perfect square?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a\\right)}^{2}\\hfill \\\\ &amp; \\text{Write it as a square.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\\\ &amp; \\text{Is the last term a perfect square?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}\\phantom{\\rule{4.5em}{0ex}}{\\left(b\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a\\right)}^{2}\\phantom{\\rule{4.5em}{0ex}}{\\left(b\\right)}^{2}\\hfill \\\\ &amp; \\text{Write it as a square.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\\\ &amp; \\text{Check the middle term. Is it}\\phantom{\\rule{0.2em}{0ex}}2ab?\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}{}_{\\text{\u2198}}\\underset{2\u00b7a\u00b7b}{}{}_{\\text{\u2199}}{\\left(b\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a\\right)}^{2}{}_{\\text{\u2198}}\\underset{2\u00b7a\u00b7b}{}{}_{\\text{\u2199}}{\\left(b\\right)}^{2}\\hfill \\\\ \\text{Step 2.}\\hfill &amp; \\text{Write the square of the binomial.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a+b\\right)}^{2}\\hfill &amp; &amp; &amp; \\hfill {\\left(a-b\\right)}^{2}\\hfill \\\\ \\text{Step 3.}\\hfill &amp; \\text{Check by multiplying.}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; \\end{array}\\)<\/li><li><strong data-effect=\"bold\">Difference of Squares Pattern:<\/strong> If \\(a,b\\) are real numbers,<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167830701187\" data-alt=\"a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.\"><\/span><\/li><li><strong data-effect=\"bold\">How to factor differences of squares.<\/strong><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{ccccc}\\text{Step 1.}\\hfill &amp; \\text{Does the binomial fit the pattern?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{a}^{2}-{b}^{2}\\hfill \\\\ &amp; \\text{Is this a difference?}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\text{____}-\\text{____}\\hfill \\\\ &amp; \\text{Are the first and last terms perfect squares?}\\hfill &amp; &amp; &amp; \\\\ \\text{Step 2.}\\hfill &amp; \\text{Write them as squares.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}{\\left(a\\right)}^{2}-{\\left(b\\right)}^{2}\\hfill \\\\ \\text{Step 3.}\\hfill &amp; \\text{Write the product of conjugates.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\left(a-b\\right)\\left(a+b\\right)\\hfill \\\\ \\text{Step 4.}\\hfill &amp; \\text{Check by multiplying.}\\hfill &amp; &amp; &amp; \\end{array}\\)<\/li><li><strong data-effect=\"bold\">Sum and Difference of Cubes Pattern<\/strong><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{c}\\hfill {a}^{3}+{b}^{3}=\\left(a+b\\right)\\left({a}^{2}-ab+{b}^{2}\\right)\\hfill \\\\ \\hfill {a}^{3}-{b}^{3}=\\left(a-b\\right)\\left({a}^{2}+ab+{b}^{2}\\right)\\hfill \\end{array}\\)<\/li><li><strong data-effect=\"bold\">How to factor the sum or difference of cubes.<\/strong><ol id=\"fs-id1167830769708\" type=\"1\" class=\"stepwise\"><li>Does the binomial fit the sum or difference of cubes pattern?<div data-type=\"newline\"><br><\/div>Is it a sum or difference?<div data-type=\"newline\"><br><\/div>Are the first and last terms perfect cubes?<\/li><li>Write them as cubes.<\/li><li>Use either the sum or difference of cubes pattern.<\/li><li>Simplify inside the parentheses<\/li><li>Check by multiplying the factors.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835371053\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167835371057\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167835371064\"><strong data-effect=\"bold\">Factor Perfect Square Trinomials<\/strong><\/p><p id=\"fs-id1167832056610\">In the following exercises, factor completely using the perfect square trinomials pattern.<\/p><div data-type=\"exercise\" id=\"fs-id1167832056614\"><div data-type=\"problem\" id=\"fs-id1167832056617\"><p id=\"fs-id1167834282608\">\\(16{y}^{2}+24y+9\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834063495\">\\({\\left(4y+3\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831871432\"><div data-type=\"problem\" id=\"fs-id1167831871434\"><p id=\"fs-id1167831871436\">\\(25{v}^{2}+20v+4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831872096\"><div data-type=\"problem\" id=\"fs-id1167831872098\"><p id=\"fs-id1167831894505\">\\(36{s}^{2}+84s+49\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834376359\"><p id=\"fs-id1167834357098\">\\({\\left(6s+7\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831106542\"><div data-type=\"problem\" id=\"fs-id1167831106544\"><p id=\"fs-id1167831106546\">\\(49{s}^{2}+154s+121\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832214525\"><div data-type=\"problem\" id=\"fs-id1167832214527\"><p>\\(100{x}^{2}-20x+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831919669\"><p id=\"fs-id1167834426188\">\\({\\left(10x-1\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448696\"><div data-type=\"problem\" id=\"fs-id1167834448698\"><p id=\"fs-id1167834448701\">\\(64{z}^{2}-16z+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830700797\"><div data-type=\"problem\" id=\"fs-id1167830700799\"><p id=\"fs-id1167834185751\">\\(25{n}^{2}-120n+144\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834228726\">\\({\\left(5n-12\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978453\"><div data-type=\"problem\" id=\"fs-id1167826978455\"><p id=\"fs-id1167826978457\">\\(4{p}^{2}-52p+169\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835419328\"><div data-type=\"problem\" id=\"fs-id1167835419330\"><p id=\"fs-id1167831887850\">\\(49{x}^{2}+28xy+4{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167828421298\"><p id=\"fs-id1167828421300\">\\({\\left(7x+2y\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834429403\"><div data-type=\"problem\" id=\"fs-id1167835328832\"><p id=\"fs-id1167835328834\">\\(25{r}^{2}+60rs+36{s}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835351862\"><p id=\"fs-id1167835351865\">\\(100{y}^{2}-52y+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831895344\"><p id=\"fs-id1167831895346\">\\(\\left(50y-1\\right)\\left(2y-1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830865346\"><div data-type=\"problem\" id=\"fs-id1167830865348\"><p id=\"fs-id1167830865351\">\\(64{m}^{2}-34m+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834120730\"><div data-type=\"problem\"><p id=\"fs-id1167834120734\">\\(10j{k}^{2}+80jk+160j\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834396515\"><p id=\"fs-id1167834419478\">\\(10j{\\left(k+4\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834179803\"><div data-type=\"problem\" id=\"fs-id1167834179805\"><p id=\"fs-id1167834179807\">\\(64{x}^{2}y-96xy+36y\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834131742\"><div data-type=\"problem\" id=\"fs-id1167834131744\"><p id=\"fs-id1167834131746\">\\(75{u}^{4}-30{u}^{3}v+3{u}^{2}{v}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835512152\"><p id=\"fs-id1167835512154\">\\(3{u}^{2}{\\left(5u-v\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835410277\"><div data-type=\"problem\"><p id=\"fs-id1167832211896\">\\(90{p}^{4}+300{p}^{4}q+250{p}^{2}{q}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167832006550\"><strong data-effect=\"bold\">Factor Differences of Squares<\/strong><\/p><p id=\"fs-id1167832006555\">In the following exercises, factor completely using the difference of squares pattern, if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1167832006559\"><div data-type=\"problem\" id=\"fs-id1167832006561\"><p id=\"fs-id1167835328059\">\\(25{v}^{2}-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835514103\"><p id=\"fs-id1167835514105\">\\(\\left(5v-1\\right)\\left(5v+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832042149\"><div data-type=\"problem\" id=\"fs-id1167832042151\"><p id=\"fs-id1167832042153\">\\(169{q}^{2}-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835489432\"><div data-type=\"problem\" id=\"fs-id1167835489434\"><p id=\"fs-id1167835489436\">\\(4-49{x}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834533145\"><p id=\"fs-id1167834533147\">\\(\\left(7x-2\\right)\\left(7x+2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835258483\"><div data-type=\"problem\" id=\"fs-id1167835258485\"><p id=\"fs-id1167835258488\">\\(121-25{s}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826996408\"><div data-type=\"problem\" id=\"fs-id1167826996410\"><p id=\"fs-id1167835629266\">\\(6{p}^{2}{q}^{2}-54{p}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832151088\"><p id=\"fs-id1167832151090\">\\(6{p}^{2}\\left(q-3\\right)\\left(q+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835420249\"><div data-type=\"problem\" id=\"fs-id1167835420251\"><p>\\(98{r}^{3}-72r\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832059445\"><div data-type=\"problem\" id=\"fs-id1167832059447\"><p id=\"fs-id1167832059449\">\\(24{p}^{2}+54\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830865848\"><p id=\"fs-id1167834489216\">\\(6\\left(4{p}^{2}+9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834472427\"><div data-type=\"problem\" id=\"fs-id1167834397195\"><p id=\"fs-id1167834397197\">\\(20{b}^{2}+140\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834430179\"><div data-type=\"problem\" id=\"fs-id1167834430182\"><p id=\"fs-id1167834430184\">\\(121{x}^{2}-144{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831846725\"><p id=\"fs-id1167831846727\">\\(\\left(11x-12y\\right)\\left(11x+12y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063544\"><div data-type=\"problem\" id=\"fs-id1167834063546\"><p id=\"fs-id1167834063548\">\\(49{x}^{2}-81{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834382416\"><div data-type=\"problem\" id=\"fs-id1167834382418\"><p id=\"fs-id1167834382420\">\\(169{c}^{2}-36{d}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827943898\"><p id=\"fs-id1167827943901\">\\(\\left(13c-6d\\right)\\left(13c+6d\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511779\"><div data-type=\"problem\" id=\"fs-id1167835511781\"><p id=\"fs-id1167835511783\">\\(36{p}^{2}-49{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830964523\"><div data-type=\"problem\" id=\"fs-id1167834387520\"><p id=\"fs-id1167834387522\">\\(16{z}^{4}-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834464139\"><p id=\"fs-id1167834464141\">\\(\\left(2z-1\\right)\\left(2z+1\\right)\\left(4{z}^{2}+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826870097\"><div data-type=\"problem\" id=\"fs-id1167826870099\"><p id=\"fs-id1167826870101\">\\({m}^{4}-{n}^{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834066085\"><div data-type=\"problem\" id=\"fs-id1167834066087\"><p id=\"fs-id1167834066089\">\\(162{a}^{4}{b}^{2}-32{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826779061\"><p id=\"fs-id1167826779063\">\\(2{b}^{2}\\left(3a-2\\right)\\left(3a+2\\right)\\left(9{a}^{2}+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832057284\"><div data-type=\"problem\" id=\"fs-id1167831112600\"><p id=\"fs-id1167831112602\">\\(48{m}^{4}{n}^{2}-243{n}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832220574\"><div data-type=\"problem\" id=\"fs-id1167832220576\"><p id=\"fs-id1167832220578\">\\({x}^{2}-16x+64-{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835346291\"><p id=\"fs-id1167835346293\">\\(\\left(x-8-y\\right)\\left(x-8+y\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831852240\"><div data-type=\"problem\" id=\"fs-id1167831852243\"><p id=\"fs-id1167831852245\">\\({p}^{2}+14p+49-{q}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835518428\"><div data-type=\"problem\" id=\"fs-id1167835518430\"><p id=\"fs-id1167835518432\">\\({a}^{2}+6a+9-9{b}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831040672\"><p id=\"fs-id1167831040675\">\\(\\left(a+3-3b\\right)\\left(a+3+3b\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167832058576\"><p id=\"fs-id1167832058578\">\\({m}^{2}-6m+9-16{n}^{2}\\)<\/p><\/div><\/div><p id=\"fs-id1167835531568\"><strong data-effect=\"bold\">Factor Sums and Differences of Cubes<\/strong><\/p><p id=\"fs-id1167835531574\">In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.<\/p><div data-type=\"exercise\" id=\"fs-id1167835531578\"><div data-type=\"problem\" id=\"fs-id1167835531580\"><p id=\"fs-id1167835531582\">\\({x}^{3}+125\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831892971\"><p id=\"fs-id1167831892973\">\\(\\left(x+5\\right)\\left({x}^{2}-5x+25\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834059830\"><div data-type=\"problem\" id=\"fs-id1167834059833\"><p id=\"fs-id1167834059835\">\\({n}^{6}+512\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826873686\"><div data-type=\"problem\"><p id=\"fs-id1167834284123\">\\({z}^{6}-27\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834284138\"><p id=\"fs-id1167831112415\">\\(\\left({z}^{2}-3\\right)\\left({z}^{4}+3{z}^{2}+9\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826781270\"><div data-type=\"problem\" id=\"fs-id1167826781272\"><p id=\"fs-id1167834228054\">\\({v}^{3}-216\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978324\"><div data-type=\"problem\" id=\"fs-id1167826978326\"><p id=\"fs-id1167826978328\">\\(8-343{t}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830702923\"><p id=\"fs-id1167830702925\">\\(\\left(2-7t\\right)\\left(4+14t+49{t}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831894684\"><div data-type=\"problem\" id=\"fs-id1167831894686\"><p id=\"fs-id1167831894688\">\\(125-27{w}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835216068\"><div data-type=\"problem\" id=\"fs-id1167835216070\"><p id=\"fs-id1167835216072\">\\(8{y}^{3}-125{z}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835530885\"><p id=\"fs-id1167835530887\">\\(\\left(2y-5z\\right)\\left(4{y}^{2}+10yz+25{z}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828447087\"><div data-type=\"problem\" id=\"fs-id1167828447089\"><p id=\"fs-id1167828447091\">\\(27{x}^{3}-64{y}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831832556\"><div data-type=\"problem\" id=\"fs-id1167832092252\"><p>\\(216{a}^{3}+125{b}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832092276\"><p id=\"fs-id1167832092278\">\\(\\left(6a+5b\\right)\\left(36{a}^{2}-30ab+25{b}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834098812\"><div data-type=\"problem\" id=\"fs-id1167834111477\"><p id=\"fs-id1167834111479\">\\(27{y}^{3}+8{z}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834562510\"><div data-type=\"problem\" id=\"fs-id1167835410116\"><p id=\"fs-id1167835410118\">\\(7{k}^{3}+56\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835410135\"><p id=\"fs-id1167835410137\">\\(7\\left(k+2\\right)\\left({k}^{2}-2k+4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834556928\"><div data-type=\"problem\" id=\"fs-id1167834556931\"><p id=\"fs-id1167834556933\">\\(6{x}^{3}-48{y}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831883770\"><div data-type=\"problem\" id=\"fs-id1167831883772\"><p id=\"fs-id1167831883774\">\\(2{x}^{2}-16{x}^{2}{y}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835517793\"><p id=\"fs-id1167835517795\">\\(2{x}^{2}\\left(1-2y\\right)\\left(1+2y+4{y}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835325286\"><div data-type=\"problem\" id=\"fs-id1167835325288\"><p id=\"fs-id1167835325290\">\\(-2{x}^{3}{y}^{2}-16{y}^{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834373428\"><div data-type=\"problem\" id=\"fs-id1167834373430\"><p id=\"fs-id1167834373432\">\\({\\left(x+3\\right)}^{3}+8{x}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834289482\"><p id=\"fs-id1167834289484\">\\(9\\left(x+1\\right)\\left({x}^{2}+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831071390\"><div data-type=\"problem\" id=\"fs-id1167831071392\"><p id=\"fs-id1167831071394\">\\({\\left(x+4\\right)}^{3}-27{x}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535070\"><div data-type=\"problem\" id=\"fs-id1167834535072\"><p id=\"fs-id1167834535074\">\\({\\left(y-5\\right)}^{3}-64{y}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835166662\"><p id=\"fs-id1167835166664\">\\(\\text{\u2212}\\left(3y+5\\right)\\left(21{y}^{2}-30y+25\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832067655\"><div data-type=\"problem\" id=\"fs-id1167832067657\"><p id=\"fs-id1167832067659\">\\({\\left(y-5\\right)}^{3}+125{y}^{3}\\)<\/p><\/div><\/div><p id=\"fs-id1167831116304\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p><p id=\"fs-id1167831116311\">In the following exercises, factor completely.<\/p><div data-type=\"exercise\" id=\"fs-id1167831116314\"><div data-type=\"problem\" id=\"fs-id1167831116316\"><p id=\"fs-id1167831116318\">\\(64{a}^{2}-25\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835390217\"><p id=\"fs-id1167835390220\">\\(\\left(8a-5\\right)\\left(8a+5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831958100\"><div data-type=\"problem\" id=\"fs-id1167831958102\"><p id=\"fs-id1167831958104\">\\(121{x}^{2}-144\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835509468\"><div data-type=\"problem\" id=\"fs-id1167835509470\"><p id=\"fs-id1167835509472\">\\(27{q}^{2}-3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835509489\"><p id=\"fs-id1167835509491\">\\(3\\left(3q-1\\right)\\left(3q+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835357654\"><div data-type=\"problem\" id=\"fs-id1167835357656\"><p id=\"fs-id1167835357659\">\\(4{p}^{2}-100\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831921204\"><div data-type=\"problem\" id=\"fs-id1167831921206\"><p id=\"fs-id1167831921208\">\\(16{x}^{2}-72x+81\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831921231\"><p id=\"fs-id1167831948996\">\\({\\left(4x-9\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831949022\"><div data-type=\"problem\" id=\"fs-id1167831949024\"><p>\\(36{y}^{2}+12y+1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621469\"><div data-type=\"problem\" id=\"fs-id1167835621471\"><p id=\"fs-id1167835621474\">\\(8{p}^{2}+2\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167830964433\">\\(2\\left(4{p}^{2}+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826873436\"><div data-type=\"problem\" id=\"fs-id1167826873438\"><p id=\"fs-id1167826873441\">\\(81{x}^{2}+169\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835510252\"><div data-type=\"problem\" id=\"fs-id1167835510255\"><p id=\"fs-id1167835510257\">\\(125-8{y}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835510274\"><p id=\"fs-id1167835510276\">\\(\\left(5-2y\\right)\\left(25+10y+4{y}^{2}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826864191\"><div data-type=\"problem\" id=\"fs-id1167826864193\"><p id=\"fs-id1167826864196\">\\(27{u}^{3}+1000\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830700659\"><div data-type=\"problem\" id=\"fs-id1167830700661\"><p>\\(45{n}^{2}+60n+20\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832066083\"><p id=\"fs-id1167832066086\">\\(5{\\left(3n+2\\right)}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826873718\"><div data-type=\"problem\" id=\"fs-id1167826873720\"><p id=\"fs-id1167826873723\">\\(48{q}^{3}-24{q}^{2}+3q\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826799330\"><div data-type=\"problem\" id=\"fs-id1167826799332\"><p id=\"fs-id1167826799335\">\\({x}^{2}-10x+25-{y}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835513292\"><p id=\"fs-id1167835513295\">\\(\\left(x+y-5\\right)\\left(x-y-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835319121\"><div data-type=\"problem\" id=\"fs-id1167835319123\"><p id=\"fs-id1167835319126\">\\({x}^{2}+12x+36-{y}^{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832152607\"><div data-type=\"problem\" id=\"fs-id1167832152609\"><p id=\"fs-id1167832152611\">\\({\\left(x+1\\right)}^{3}+8{x}^{3}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835357746\"><p id=\"fs-id1167835357748\">\\(\\left(3x+1\\right)\\left(3{x}^{2}+1\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167828426665\"><div data-type=\"problem\" id=\"fs-id1167828426667\"><p id=\"fs-id1167828426669\">\\({\\left(y-3\\right)}^{3}-64{y}^{3}\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167831872220\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167834394383\"><div data-type=\"problem\" id=\"fs-id1167834394385\"><p id=\"fs-id1167834394387\">Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834394393\"><p>Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834394400\"><div data-type=\"problem\" id=\"fs-id1167834394402\"><p id=\"fs-id1167834394404\">How do you recognize the binomial squares pattern?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831117390\"><div data-type=\"problem\" id=\"fs-id1167831117393\"><p id=\"fs-id1167831117395\">Explain why \\({n}^{2}+25\\ne {\\left(n+5\\right)}^{2}.\\) Use algebra, words, or pictures.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834525248\"><p id=\"fs-id1167834525250\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834525255\"><div data-type=\"problem\" id=\"fs-id1167834525257\"><p id=\"fs-id1167834525259\">Maribel factored \\({y}^{2}-30y+81\\) as \\({\\left(y-9\\right)}^{2}.\\) Was she right or wrong? How do you know?<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832151236\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167832151241\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167831919620\" data-alt=\"This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.\"><\/span><p id=\"fs-id1167831919631\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p><\/div><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Factor perfect square trinomials<\/li>\n<li>Factor differences of squares<\/li>\n<li>Factor sums and differences of cubes<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" class=\"be-prepared\">\n<p>Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835280232\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6aaa6bb05ea5df8c4e42d93be998f10e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"53\" style=\"vertical-align: -7px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/3fa6a6c5-9a36-4dee-aea1-0166229f52fb#fs-id1167835304261\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccfbc543b8925ab065b9fa99a64af55d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#109;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"71\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6160a75809166152b48fe6e23fc9ff6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"118\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836717042\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167835423012\">We have seen that some binomials and trinomials result from special products\u2014squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167830769807\">\n<h3 data-type=\"title\">Factor Perfect Square Trinomials<\/h3>\n<p>Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter.<\/p>\n<p><span data-type=\"media\" data-alt=\"In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.\" \/><\/span><\/p>\n<p>The trinomial <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ed5c09832c27302ab19f40b4cecc7db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#120;&#43;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -2px;\" \/> is called a <em data-effect=\"italics\">perfect square trinomial<\/em>. It is the square of the binomial <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b36107b910e40f1a5c35d39c6537008_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"54\" style=\"vertical-align: -2px;\" \/><\/p>\n<p id=\"fs-id1167835367325\">In this chapter, you will start with a perfect square trinomial and factor it into its <span data-type=\"term\" class=\"no-emphasis\">prime<\/span> factors.<\/p>\n<p>You could factor this <span data-type=\"term\" class=\"no-emphasis\">trinomial<\/span> using the methods described in the last section, since it is of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13fc8234342051cf7b36d52613e9e1f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#98;&#120;&#43;&#99;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -2px;\" \/> But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.<\/p>\n<p id=\"fs-id1167834120157\">Here is the pattern\u2014the reverse of the binomial squares pattern.<\/p>\n<div data-type=\"note\" id=\"fs-id1167832076542\">\n<div data-type=\"title\">Perfect Square Trinomials Pattern<\/div>\n<p id=\"fs-id1167832153960\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834535644\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8fecc189c4ae314308f222184adcf71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"185\" style=\"vertical-align: -16px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834430982\">To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0db73d0a0f1d3892762402d8c751aaab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: 0px;\" \/> Next check that the last term is a perfect square, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cce4a9873ea8b9326082bc75fbac9081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\" \/> Then check the middle term\u2014is it the product, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95818de2c5f58f5d0a1c0e49297bb76a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;&#98;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"34\" style=\"vertical-align: 0px;\" \/> If everything checks, you can easily write the factors.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832056529\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Factor Perfect Square Trinomials<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834533424\">\n<div data-type=\"problem\" id=\"fs-id1167831911097\">\n<p id=\"fs-id1167835352469\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e59d19437e46075d0c45cae1e9c23fc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306597\"><span data-type=\"media\" id=\"fs-id1167835322243\" data-alt=\"Step 1 is to check if the trinomial fits the perfect square trinomials pattern, a squared plus 2ab plus b squared. For this we check if the first term is a perfect square. 9 x squared is the square of 3x. Next we check if the last term is a perfect square. 4 is the square of 2. Next we check if the middle term is 2ab. 12 x is twice 3x times 2. Hence we have a perfect square trinomial.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to check if the trinomial fits the perfect square trinomials pattern, a squared plus 2ab plus b squared. For this we check if the first term is a perfect square. 9 x squared is the square of 3x. Next we check if the last term is a perfect square. 4 is the square of 2. Next we check if the middle term is 2ab. 12 x is twice 3x times 2. Hence we have a perfect square trinomial.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835335949\" data-alt=\"Step 2 is to write this as the square of a binomial. We write it as open parentheses 3x plus 2 close parentheses squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to write this as the square of a binomial. We write it as open parentheses 3x plus 2 close parentheses squared.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 3 is to check by multiplying.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to check by multiplying.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167827940329\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835364113\">\n<p id=\"fs-id1167835263503\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67945c25e7d6bce21d789fc5866b763e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832082000\">\n<p id=\"fs-id1167835238962\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c61c1e6b05e1dc0893efd01f132085f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835369810\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831883553\">\n<div data-type=\"problem\" id=\"fs-id1167828421283\">\n<p id=\"fs-id1167834189852\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9099b9f5d0bf71f03fdc475f3b62f925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#121;&#43;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826857212\">\n<p id=\"fs-id1167831191480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0215b75c774aa0e0ab59bad40fec14d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835417985\">The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3f521bdd8891f6825dfedeaef27d0d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -4px;\" \/> which factors to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4f293a361a450256c7e79911902c001_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167832068338\">The steps are summarized here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834505010\" class=\"howto\">\n<div data-type=\"title\">Factor perfect square trinomials.<\/div>\n<p id=\"fs-id1167834213922\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f203b5713fe6cc23aee261c6fbee3824_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#102;&#105;&#116;&#32;&#116;&#104;&#101;&#32;&#112;&#97;&#116;&#116;&#101;&#114;&#110;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#32;&#97;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#116;&#32;&#97;&#115;&#32;&#97;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#32;&#97;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#116;&#32;&#97;&#115;&#32;&#97;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#116;&#104;&#101;&#32;&#109;&#105;&#100;&#100;&#108;&#101;&#32;&#116;&#101;&#114;&#109;&#46;&#32;&#73;&#115;&#32;&#105;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#97;&#98;&#63;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8600;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#50;&middot;&#97;&middot;&#98;&#125;&#123;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8601;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8600;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#50;&middot;&#97;&middot;&#98;&#125;&#123;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8601;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"184\" width=\"787\" style=\"vertical-align: -86px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167835377433\">We\u2019ll work one now where the middle term is negative.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835345249\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834382630\">\n<div data-type=\"problem\" id=\"fs-id1167831811607\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d24d8c397d5685cbd9d0a7bb2c5f43cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#50;&#121;&#43;&#49;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834194750\">\n<p id=\"fs-id1167834229171\">The first and last terms are squares. See if the middle term fits the pattern of a <span data-type=\"term\" class=\"no-emphasis\">perfect square<\/span> trinomial. The middle term is negative, so the binomial square would be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4f293a361a450256c7e79911902c001_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/p>\n<table class=\"unnumbered unstyled\" summary=\"The trinomial is 81 y squared minus 72y plus 16. The first and last terms are perfect squares of 9y and 4. Check the middle term. It is twice 9y times 4. The trinomial matches a minus b the whole squared. So we write it as the square of a binomial open parentheses 9y minus 4 close parentheses squared. Finally, we check by multiplying.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832053914\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Are the first and last terms perfect squares?\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834301208\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check the middle term.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831985779\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Does it match <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e077a0afbd98583e11e87f2eb25bd89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"67\" style=\"vertical-align: -4px;\" \/> Yes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834130259\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write as the square of a binomial.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228768\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check by multiplying:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d430b6e77086e40df3a50abf08803b3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&middot;&#57;&#121;&middot;&#52;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#50;&#121;&#43;&#49;&#54;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"134\" style=\"vertical-align: -27px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835257249\">\n<div data-type=\"problem\" id=\"fs-id1167831928925\">\n<p id=\"fs-id1167835258025\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad356f0c1d2669e31a33d7dab36939c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#48;&#121;&#43;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"127\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834505103\">\n<p id=\"fs-id1167834189773\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67d229cf541d2d34e6d1e0e1c1f94f54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835235994\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832152848\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831882541\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d75bb70666eb7a44de576b595057b00b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#50;&#122;&#43;&#56;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835361589\">\n<p id=\"fs-id1167832044141\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8d213e0f3c101304d0209d4fec44e6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#122;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835330168\">The next example will be a perfect square trinomial with two variables.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835254902\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835193710\">\n<div data-type=\"problem\" id=\"fs-id1167834061377\">\n<p id=\"fs-id1167835274949\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cb444e951267b9f19932302e71c1c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#52;&#120;&#121;&#43;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832052324\">\n<table id=\"fs-id1167835353428\" class=\"unnumbered unstyled\" summary=\"The trinomial is 36 x squared plus 84xy plus 49 y squared. We test each term to verify the pattern. The trinomial is open parentheses 6x close parentheses squared plus 2 times 6x times 7y plus open parentheses 7y close parentheses squared. We factor to get open parentheses 6x plus 7y close parentheses squared.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834432988\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Test each term to verify the pattern.\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835338542\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835233638\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check by multiplying.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93d73bc8bed669a7f1158a7024e55bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#43;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&middot;&#54;&#120;&middot;&#55;&#121;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#52;&#120;&#121;&#43;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"168\" style=\"vertical-align: -27px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834301082\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835231699\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c922cfadb5e67e9f1bc9b50e0745d540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#52;&#120;&#121;&#43;&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"155\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835344675\">\n<p id=\"fs-id1167832152791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f2cbf2d226ca8e0ecbc988b48bd0492a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#120;&#43;&#54;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835345375\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832057915\">\n<div data-type=\"problem\" id=\"fs-id1167831893298\">\n<p id=\"fs-id1167835375915\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2df7e5352de0572c95a15c9d754c919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#49;&#50;&#109;&#110;&#43;&#52;&#57;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"177\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834372379\">\n<p id=\"fs-id1167832152694\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc2936d4122d061c0f916ceb0490b9d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#109;&#43;&#55;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"85\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835357616\">Remember the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a <span data-type=\"term\" class=\"no-emphasis\">GCF<\/span> in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834396304\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835370069\">\n<div data-type=\"problem\" id=\"fs-id1167831882219\">\n<p id=\"fs-id1167834065318\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5afadfee258a9b568177e9be9ca42a39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#56;&#48;&#120;&#121;&#43;&#49;&#54;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"164\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167835513206\" class=\"unnumbered unstyled\" summary=\"Is there a GCF in 100 x squared y minus 80xy plus 16y? Yes. Factoring it out, we get 4y open parentheses 25 x squared minus 20x plus 4 close parentheses. Is this a perfect square trinomial? To verify the pattern, we rewrite as 4y open bracket open parentheses 5x close parentheses squared minus 2 times 5x times 2 plus 2 squared close bracket. Factor to get 4y open parentheses 5x minus 2 close parentheses squared. Finally, we check by multiplying.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834196287\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Is there a GCF? Yes, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb2c2e54f255146a838bfcf255041029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#121;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"22\" style=\"vertical-align: -4px;\" \/> so factor it out.\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832152888\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Is this a perfect square trinomial?<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Verify the pattern.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834329838\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember: Keep the factor 4<em data-effect=\"italics\">y<\/em> in the final product.<\/p>\n<p id=\"fs-id1167832086884\">Check:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac26bb8552a61652512c7e4929a7f468_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#121;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#121;&#92;&#108;&#101;&#102;&#116;&#91;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&middot;&#53;&#120;&middot;&#50;&#43;&#123;&#50;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#121;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#56;&#48;&#120;&#121;&#43;&#49;&#54;&#121;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"172\" style=\"vertical-align: -42px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835348534\">\n<div data-type=\"problem\" id=\"fs-id1167834557137\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daf1cb0a12d334f5c575362f73b942bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#50;&#52;&#120;&#121;&#43;&#49;&#56;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834193624\">\n<p id=\"fs-id1167832075544\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f995a50098ad5af86ed95888b45470ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#121;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167830705949\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7da703dcacb94c34f08fb0a4d770cba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#113;&#43;&#57;&#48;&#112;&#113;&#43;&#55;&#53;&#113;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835349497\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c091e37500fe67d4e690f26e293d7b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#113;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#112;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Factor Differences of Squares<\/h3>\n<p id=\"fs-id1167834185224\">The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here\u2019s an example:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826987977\" data-alt=\"We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_006_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.\" \/><\/span><\/p>\n<p>A difference of squares factors to a product of conjugates.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834063594\">\n<div data-type=\"title\">Difference of Squares Pattern<\/div>\n<p id=\"fs-id1167834124313\">If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,<\/p>\n<p><span data-type=\"media\" data-alt=\"a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are conjugates.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are conjugates.\" \/><\/span><\/div>\n<p id=\"fs-id1167826857435\">Remember, \u201cdifference\u201d refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Factor a Trinomial Using the Difference of Squares<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835326544\">\n<p>Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da82653edcc488b639c11468c20a860b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832226786\"><span data-type=\"media\" id=\"fs-id1167835344526\" data-alt=\"Step 1 is to check if the binomial 64 y squared minus 1 fits the pattern. For that we check the following: Is this a difference? Yes. Are the first and last terms perfect squares? Yes.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to check if the binomial 64 y squared minus 1 fits the pattern. For that we check the following: Is this a difference? Yes. Are the first and last terms perfect squares? Yes.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835417524\" data-alt=\"Step 2 is to write both terms as squares, So, we have open parentheses 8y close parentheses squared minus 1 squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to write both terms as squares, So, we have open parentheses 8y close parentheses squared minus 1 squared.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835380209\" data-alt=\"Step 3 is to write the product of conjugates 8y minus 1, 8y plus 1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to write the product of conjugates 8y minus 1, 8y plus 1.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167832075154\" data-alt=\"Step 4 is to check. We multiply to get the original binomial\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_008d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to check. We multiply to get the original binomial\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835596424\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834130741\">\n<div data-type=\"problem\" id=\"fs-id1167835303608\">\n<p id=\"fs-id1167834085029\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c11a0599609ed851a7e283b85c54764_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831112564\">\n<p id=\"fs-id1167834463142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8836411ca6c167fae53d826ead9d6c26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#109;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#109;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834429753\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834522827\">\n<div data-type=\"problem\" id=\"fs-id1167835324763\">\n<p id=\"fs-id1167834324728\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e2ac53fc80142b2e5ae5c06f88b4073_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420022\">\n<p id=\"fs-id1167835299916\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19fa4d5aa8bf16b7002b73840d52ca77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835369387\" class=\"howto\">\n<div data-type=\"title\">Factor differences of squares.<\/div>\n<p id=\"fs-id1167831922306\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#102;&#105;&#116;&#32;&#116;&#104;&#101;&#32;&#112;&#97;&#116;&#116;&#101;&#114;&#110;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#105;&#115;&#32;&#97;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#97;&#110;&#100;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#109;&#32;&#97;&#115;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;\r\n\n<\/pre>\n<\/div>\n<p id=\"fs-id1167831116937\">It is important to remember that <em data-effect=\"italics\">sums of squares do not factor into a product of binomials<\/em>. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84264a6e3af632c1d67d4907fa750268_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -2px;\" \/> is prime!<\/p>\n<p id=\"fs-id1171791716465\">The next example shows variables in both terms.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834228380\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835257523\">\n<div data-type=\"problem\" id=\"fs-id1167834228385\">\n<p id=\"fs-id1167832042469\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5b2ccee7a3be4f5987fb70ba9a20c0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831944012\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86c24090274ff7ad68d504a080106eb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#105;&#115;&#32;&#97;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#63;&#32;&#89;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#45;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#43;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#45;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#120;&#43;&#55;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#52;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"175\" width=\"568\" style=\"vertical-align: -82px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834397185\">\n<div data-type=\"problem\" id=\"fs-id1167832058472\">\n<p id=\"fs-id1167835371022\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3640a8422a76cb183e44d325037da1ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#57;&#54;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"111\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834162074\">\n<p id=\"fs-id1167834526533\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb2c6aee1d6f7742d3e66cb095835d89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#109;&#45;&#53;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#109;&#43;&#53;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"178\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835349317\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831103990\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835171108\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea9e4a3af86c36bb621e85eb815449ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835381273\">\n<p id=\"fs-id1167835308295\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9333f84eab6de90a0c6dc2ddfeabadbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#112;&#45;&#51;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#112;&#43;&#51;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"160\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834300692\">As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may \u201cdisguise\u201d the difference of squares and you won\u2019t recognize the perfect squares until you factor the GCF.<\/p>\n<p id=\"fs-id1167826819804\">Also, to completely factor the binomial in the next example, we\u2019ll factor a difference of squares twice!<\/p>\n<div data-type=\"example\" id=\"fs-id1167832042561\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835514442\">\n<div data-type=\"problem\" id=\"fs-id1167826880317\">\n<p id=\"fs-id1167835317851\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-604acdbb4442e3b4b11d900999465d6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831871375\">\n<p id=\"fs-id1167834064508\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fd11c58902b3c3cad77302bde1066eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#56;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#114;&#101;&#32;&#97;&#32;&#71;&#67;&#70;&#63;&#32;&#89;&#101;&#115;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8212;&#102;&#97;&#99;&#116;&#111;&#114;&#32;&#105;&#116;&#32;&#111;&#117;&#116;&#33;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#56;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#97;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#63;&#32;&#89;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#97;&#115;&#32;&#97;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#111;&#116;&#105;&#99;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#105;&#115;&#32;&#97;&#108;&#115;&#111;&#32;&#97;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#33;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#97;&#99;&#116;&#111;&#114;&#32;&#105;&#116;&#32;&#97;&#115;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"152\" width=\"739\" style=\"vertical-align: -72px;\" \/><\/p>\n<p id=\"fs-id1167832226903\">The last factor, the sum of squares, cannot be factored.<\/p>\n<p id=\"fs-id1167832015899\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3048cc27688d7138366c59c5102c8836_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#45;&#56;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#56;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#10003;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"149\" width=\"304\" style=\"vertical-align: -70px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834423133\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834423136\">\n<div data-type=\"problem\" id=\"fs-id1167835361669\">\n<p id=\"fs-id1167835361671\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a67babe377e704e02cc1abab9a931ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#52;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830700962\">\n<p id=\"fs-id1167830700964\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b25000cf681d26c26f4a0e586552277d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"206\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835374550\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835174151\">\n<div data-type=\"problem\" id=\"fs-id1167835420262\">\n<p id=\"fs-id1167835420264\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14b2627a3f82eb9acb8982a06e84d529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#123;&#98;&#125;&#94;&#123;&#52;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"106\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830960797\">\n<p id=\"fs-id1167830960799\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08632a0ef7f5ddbeac50678bda8da8a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"206\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835288115\">The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835377553\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831894420\">\n<div data-type=\"problem\" id=\"fs-id1167831894422\">\n<p id=\"fs-id1167834532547\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc8097b4bed9cb8531e2b1869d7cba8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#43;&#57;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831910960\">\n<p id=\"fs-id1167826778526\">Notice that the first three terms form a perfect square trinomial.<\/p>\n<table id=\"fs-id1167832076022\" class=\"unnumbered unstyled\" summary=\"We have x squared minus 6x plus 9 minus y squared. We factor by grouping the first three terms. Use the perfect square trinomial pattern to get open parentheses x minus 3 close parentheses squared minus y squared. Is this a difference of squares? Yes. Factor as the product of conjugates, open parentheses x minus 3 minus y close parentheses open parentheses x minus 3 plus y close parentheses. You may want to rewrite the solution as open parentheses x minus y minus 3 close parentheses open parentheses x plus y minus 3 close parentheses.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835310550\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor by grouping the first three terms.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835339144\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the perfect square trinomial pattern.\u2003\u2003\u2003\u2003\u2003<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167832211979\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Is this a difference of squares? Yes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Yes\u2014write them as squares.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167826802852\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor as the product of conjugates.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830865387\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834473752\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834299522\">You may want to rewrite the solution as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19173c47083484cadd40f9b59ff6adba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"180\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834213904\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832041469\">\n<div data-type=\"problem\" id=\"fs-id1167832041472\">\n<p id=\"fs-id1167832041474\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb50131f16cdab67b2d8abc0befa725c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#50;&#53;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835329671\">\n<p id=\"fs-id1167835329674\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-507a49326805b2976ecb916876b0e827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#53;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834340149\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167828434986\">\n<div data-type=\"problem\" id=\"fs-id1167828434988\">\n<p id=\"fs-id1167835318775\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2acbe941d57b7d930f6a0b816656051_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#120;&#43;&#57;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832036135\">\n<p id=\"fs-id1167832036137\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b46514a5655ab70519530bef36fd9a9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#43;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"190\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834137540\">\n<h3 data-type=\"title\">Factor Sums and Differences of Cubes<\/h3>\n<p id=\"fs-id1167834137546\">There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835410207\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-209caa26a891520c477f883d2693b5ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"240\" style=\"vertical-align: -18px;\" \/><\/div>\n<p id=\"fs-id1167834222369\">We\u2019ll check the first pattern and leave the second to you.<\/p>\n<table id=\"fs-id1167835303688\" class=\"unnumbered unstyled\" summary=\"Open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared. Distribute: a open parentheses a squared minus ab plus b squared plus b open parentheses a squared minus ab plus b squared close parentheses. Multiply: a cubed minus a squared b plus ab squared plus a squared b minus ab squared plus b cubed. Combine like terms: a cubed plus b cubed.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826937760\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Distribute.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834517541\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Combine like terms.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835421924\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"note\" id=\"fs-id1167835353807\">\n<div data-type=\"title\">Sum and Difference of Cubes Pattern<\/div>\n<div data-type=\"equation\" id=\"fs-id1167830865652\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-209caa26a891520c477f883d2693b5ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"240\" style=\"vertical-align: -18px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835483694\">The two patterns look very similar, don\u2019t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835483696\" data-alt=\"a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.\" \/><\/span><\/p>\n<p id=\"fs-id1167834133012\">The trinomial factor in the sum and difference of cubes pattern cannot be factored.<\/p>\n<p id=\"fs-id1167834133016\">It be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in <a href=\"#fs-id1167835337696\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"fs-id1167835337696\" summary=\"This table has 11 columns and 2 columns. The first column labels each row n and n cubed. The remaining columns of the first row have the numbers 1 through 10. The remaining columns of the second row have the numbers 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">n<\/em><\/th>\n<th data-valign=\"top\" data-align=\"left\">1<\/th>\n<th data-valign=\"top\" data-align=\"left\">2<\/th>\n<th data-valign=\"top\" data-align=\"left\">3<\/th>\n<th data-valign=\"top\" data-align=\"left\">4<\/th>\n<th data-valign=\"top\" data-align=\"left\">5<\/th>\n<th data-valign=\"top\" data-align=\"left\">6<\/th>\n<th data-valign=\"top\" data-align=\"left\">7<\/th>\n<th data-valign=\"top\" data-align=\"left\">8<\/th>\n<th data-valign=\"top\" data-align=\"left\">9<\/th>\n<th data-valign=\"top\" data-align=\"left\">10<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f6e7910f74eb6b19c4999a19bfc21bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\">1<\/td>\n<td data-valign=\"top\" data-align=\"left\">8<\/td>\n<td data-valign=\"top\" data-align=\"left\">27<\/td>\n<td data-valign=\"top\" data-align=\"left\">64<\/td>\n<td data-valign=\"top\" data-align=\"left\">125<\/td>\n<td data-valign=\"top\" data-align=\"left\">216<\/td>\n<td data-valign=\"top\" data-align=\"left\">343<\/td>\n<td data-valign=\"top\" data-align=\"left\">512<\/td>\n<td data-valign=\"top\" data-align=\"left\">729<\/td>\n<td data-valign=\"top\" data-align=\"left\">1000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1167830963362\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Factor the Sum or Difference of Cubes<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830963365\">\n<div data-type=\"problem\" id=\"fs-id1167830963367\">\n<p id=\"fs-id1167830963369\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c87b38a4b36f99821de271cc05c036d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#54;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834516173\"><span data-type=\"media\" id=\"fs-id1167834516175\" data-alt=\"Step 1 is to check if the binomial fits the sum or difference of cubes pattern. For this, we check whether it is a sum or difference. x cubed plus 64 is a sum. Next we check if the first and last terms are perfect cubes. They are\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to check if the binomial fits the sum or difference of cubes pattern. For this, we check whether it is a sum or difference. x cubed plus 64 is a sum. Next we check if the first and last terms are perfect cubes. They are\" \/><\/span><span data-type=\"media\" id=\"fs-id1167826986850\" data-alt=\"Step 2 is to rewrite as cubes. So we rewrite as x cubed plus 4 cubed.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to rewrite as cubes. So we rewrite as x cubed plus 4 cubed.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167831881792\" data-alt=\"Step 3 is to use either the sum or difference of cubes pattern. Since this is a sum of cubes, we get open parentheses x plus 4 close parentheses open parentheses x squared minus 4x plus 4 squared.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to use either the sum or difference of cubes pattern. Since this is a sum of cubes, we get open parentheses x plus 4 close parentheses open parentheses x squared minus 4x plus 4 squared.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834367159\" data-alt=\"Step 4 is to simplify inside the parentheses. It is already simplified\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to simplify inside the parentheses. It is already simplified\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834192367\" data-alt=\"Step 5 is to check by multiplying the factors.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_012e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to check by multiplying the factors.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826801876\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826801879\">\n<div data-type=\"problem\" id=\"fs-id1167826801881\">\n<p id=\"fs-id1167835336557\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-073d65290a89fc1c69c348c5917848c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"61\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835299809\">\n<p id=\"fs-id1167835299811\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d29dd6d42c02299c9593772490e0676e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#120;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"160\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835368092\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835368095\">\n<div data-type=\"problem\" id=\"fs-id1167831115433\">\n<p id=\"fs-id1167831115435\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d3befc7fcec3ee27d0c274be65dcd9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370061\">\n<p id=\"fs-id1167835370063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fac4c792a4afb8349651a5439ac49276_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"157\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830700252\" class=\"howto\">\n<div data-type=\"title\">Factor the sum or difference of cubes.<\/div>\n<ol id=\"fs-id1167835282990\" type=\"1\" class=\"stepwise\">\n<li>Does the binomial fit the sum or difference of cubes pattern?\n<div data-type=\"newline\"><\/div>\n<p>Is it a sum or difference?<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Are the first and last terms perfect cubes?<\/li>\n<li>Write them as cubes.<\/li>\n<li>Use either the sum or difference of cubes pattern.<\/li>\n<li>Simplify inside the parentheses.<\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835510228\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835510230\">\n<div data-type=\"problem\" id=\"fs-id1167831970070\">\n<p id=\"fs-id1167831970072\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-86011752a7b9095ffbca807db6766d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#53;&#123;&#118;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835281405\">\n<table id=\"fs-id1167835281408\" class=\"unnumbered unstyled\" summary=\"The binomial is 27 u cubed minus 125 v cubed. It is a difference. The first and last terms are perfect cubes. We rewrite as open parentheses 3u close parentheses cubed minus open parentheses 5v close parentheses cubed. Using the difference of cubes pattern, we get open parentheses 3u minus 5v close parentheses open parentheses open parentheses3u close parentheses squared plus 3u times 5v plus open parentheses 5v close parentheses squared close parentheses. We simplify to get open parentheses 3u minus 5v close parentheses open parentheses 9 u squared plus 15uv plus 25 v squared close parentheses. Finally, check by multiplying.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830706026\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">This binomial is a difference. The first and last<\/p>\n<div data-type=\"newline\"><\/div>\n<p>terms are perfect cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the terms as cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834525139\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the difference of cubes pattern.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835519133\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834396869\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check by multiplying.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We\u2019ll leave the check to you.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834345963\">\n<div data-type=\"problem\" id=\"fs-id1167834345965\">\n<p id=\"fs-id1167834345967\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6faee3a06ffef9d9b5d9b9fb802d691_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#55;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834131074\">\n<p id=\"fs-id1167834534784\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd6ff7a63b7083d4723d88efc199faff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#120;&#45;&#51;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#121;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"213\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831894757\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831040260\">\n<div data-type=\"problem\" id=\"fs-id1167831040262\">\n<p id=\"fs-id1167831040264\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20f5495f82a7b2a8a6392f7d4305fc6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#48;&#123;&#109;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#53;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"128\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832053341\">\n<p id=\"fs-id1167832053343\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d765a22d42c84b5236dac90ef7b28a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#109;&#45;&#53;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#48;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#48;&#109;&#110;&#43;&#50;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"278\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834534369\">In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834534372\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834346225\">\n<div data-type=\"problem\" id=\"fs-id1167834346227\">\n<p id=\"fs-id1167834346230\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c6e06910f73f381a0be2faad0add72f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#121;&#43;&#52;&#56;&#123;&#121;&#125;&#94;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826802403\">\n<table id=\"fs-id1167826802406\" class=\"unnumbered unstyled\" summary=\"We have 6 x cubed y plus 48 y to the power 4. Factoring the common factor, we rewrite as 6y open parentheses x cubed plus 8 y cubed close parentheses. This binomial is a sum and the first and last terms are perfect cubes. We rewrite as 6y open brackets x cubed plus open parentheses 2y close parentheses cubed close bracket. Using the sum of cubes pattern and simplifying, we get 6y open parentheses x plus 2y close parentheses open parentheses x squared minus 2xy plus 4 y squared close parentheses.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835319053\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Factor the common factor.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831921402\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">This binomial is a sum The first and last<\/p>\n<div data-type=\"newline\"><\/div>\n<p>terms are perfect cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the terms as cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835623236\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the sum of cubes pattern.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826987383\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826937937\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167832055056\">Check:<\/p>\n<p id=\"fs-id1167832055059\">To check, you may find it easier to multiply the sum of cubes factors first, then multiply that product by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-902bf03feb4b62bcebb67b67452a674c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"22\" style=\"vertical-align: -4px;\" \/> We\u2019ll leave the multiplication for you.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832010232\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832010236\">\n<div data-type=\"problem\" id=\"fs-id1167832010238\">\n<p id=\"fs-id1167832010240\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b045f4312481099610c174c43c7fff27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;&#48;&#123;&#112;&#125;&#94;&#123;&#51;&#125;&#43;&#52;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834472825\">\n<p id=\"fs-id1167834472827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4b2ec74d6f0fbb8e8498f7c6ada2a50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#112;&#43;&#113;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#53;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#112;&#113;&#43;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"211\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167828436488\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167828436492\">\n<div data-type=\"problem\" id=\"fs-id1167828436494\">\n<p id=\"fs-id1167831836142\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31786f09a923eec94d2a3e51c146ec88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#51;&#50;&#123;&#99;&#125;&#94;&#123;&#51;&#125;&#43;&#54;&#56;&#54;&#123;&#100;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834524246\">\n<p id=\"fs-id1167834524248\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f96e07e4016fe133f8231a4623547c5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#99;&#43;&#55;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#54;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#50;&#99;&#100;&#43;&#52;&#57;&#123;&#100;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"245\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1171791690075\">The first term in the next example is a binomial cubed.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835417027\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835417029\">\n<div data-type=\"problem\" id=\"fs-id1167826807776\">\n<p id=\"fs-id1167826807778\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9734d41ff39e242a7820a86ffbf5dc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832055514\">\n<table id=\"fs-id1167832055518\" class=\"unnumbered unstyled\" summary=\"The binomial open parentheses x plus 5 close parentheses cubed minus 64 x cubed is a difference. The first and last terms are perfect cubes. Using the difference of cubes pattern and simplifying, we get open parentheses minus 3x plus 5 close parentheses open parentheses 21 x squared plus 30x plus 25 close parentheses. Check by multiplying.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834438875\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">This binomial is a difference. The first and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>last terms are perfect cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the terms as cubes.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835371036\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Use the difference of cubes pattern.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831922972\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834463097\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Check by multiplying.<\/td>\n<td data-valign=\"top\" data-align=\"left\">We\u2019ll leave the check to you.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831825658\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831825663\">\n<div data-type=\"problem\" id=\"fs-id1167831825665\">\n<p id=\"fs-id1167831825667\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a47962a4a200ae951263146f099a992_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#55;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219510\">\n<p id=\"fs-id1167834219512\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d49a36abd4ec3ec7a620f23e64a2b18c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"198\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167827967115\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167827967119\">\n<div data-type=\"problem\" id=\"fs-id1167827967121\">\n<p id=\"fs-id1167831913457\">Factor: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ce69afcb236c42119783a52ed86f9b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#53;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835359626\">\n<p id=\"fs-id1167835359628\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08f0094dfccda03b1cea623760f23a20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#110;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#49;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#49;&#110;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"210\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830837043\" class=\"media-2\">\n<p id=\"fs-id1167830837047\">Access this online resource for additional instruction and practice with factoring special products.<\/p>\n<ul id=\"fs-id1167832153726\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37BinomCubes\">Factoring Binomials-Cubes #2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835414608\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835414615\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Perfect Square Trinomials Pattern:<\/strong> If <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em> are real numbers,\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167831148856\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8fecc189c4ae314308f222184adcf71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"185\" style=\"vertical-align: -16px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">How to factor perfect square trinomials.<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2acc7c0a5e5b0df678cfc182694b202c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#116;&#114;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#102;&#105;&#116;&#32;&#116;&#104;&#101;&#32;&#112;&#97;&#116;&#116;&#101;&#114;&#110;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#32;&#97;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#116;&#32;&#97;&#115;&#32;&#97;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#101;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#32;&#97;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#105;&#116;&#32;&#97;&#115;&#32;&#97;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#116;&#104;&#101;&#32;&#109;&#105;&#100;&#100;&#108;&#101;&#32;&#116;&#101;&#114;&#109;&#46;&#32;&#73;&#115;&#32;&#105;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#97;&#98;&#63;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8600;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#50;&middot;&#97;&middot;&#98;&#125;&#123;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8601;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8600;&#125;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#50;&middot;&#97;&middot;&#98;&#125;&#123;&#125;&#123;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8601;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"184\" width=\"787\" style=\"vertical-align: -86px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Difference of Squares Pattern:<\/strong> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfaed44949cf9cfbeb3445de33aabd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"25\" style=\"vertical-align: -4px;\" \/> are real numbers,\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167830701187\" data-alt=\"a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_016_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"a squared minus b squared is a minus b, a plus b. Here, a squared minus b squared is the difference of squares and a minus b, a plus b are conjugates.\" \/><\/span><\/li>\n<li><strong data-effect=\"bold\">How to factor differences of squares.<\/strong>\n<div data-type=\"newline\"><\/div>\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#49;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#101;&#115;&#32;&#116;&#104;&#101;&#32;&#98;&#105;&#110;&#111;&#109;&#105;&#97;&#108;&#32;&#102;&#105;&#116;&#32;&#116;&#104;&#101;&#32;&#112;&#97;&#116;&#116;&#101;&#114;&#110;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#115;&#32;&#116;&#104;&#105;&#115;&#32;&#97;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#97;&#110;&#100;&#32;&#108;&#97;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#112;&#101;&#114;&#102;&#101;&#99;&#116;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#63;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#50;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#109;&#32;&#97;&#115;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#51;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#112;&#114;&#111;&#100;&#117;&#99;&#116;&#32;&#111;&#102;&#32;&#99;&#111;&#110;&#106;&#117;&#103;&#97;&#116;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#116;&#101;&#112;&#32;&#52;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#104;&#101;&#99;&#107;&#32;&#98;&#121;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#105;&#110;&#103;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error 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data-effect=\"bold\">Sum and Difference of Cubes Pattern<\/strong>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-209caa26a891520c477f883d2693b5ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#98;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#45;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#97;&#98;&#43;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"240\" style=\"vertical-align: -18px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How to factor the sum or difference of cubes.<\/strong>\n<ol id=\"fs-id1167830769708\" type=\"1\" class=\"stepwise\">\n<li>Does the binomial fit the sum or difference of cubes pattern?\n<div data-type=\"newline\"><\/div>\n<p>Is it a sum or difference?<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Are the first and last terms perfect cubes?<\/li>\n<li>Write them as cubes.<\/li>\n<li>Use either the sum or difference of cubes pattern.<\/li>\n<li>Simplify inside the parentheses<\/li>\n<li>Check by multiplying the factors.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835371053\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167835371057\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167835371064\"><strong data-effect=\"bold\">Factor Perfect Square Trinomials<\/strong><\/p>\n<p id=\"fs-id1167832056610\">In the following exercises, factor completely using the perfect square trinomials pattern.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832056614\">\n<div data-type=\"problem\" id=\"fs-id1167832056617\">\n<p id=\"fs-id1167834282608\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9a601cf14468bb634ecb1072d2a09d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#52;&#121;&#43;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834063495\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b28ed2ea39b649245eb60f83a98c656_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831871432\">\n<div data-type=\"problem\" id=\"fs-id1167831871434\">\n<p id=\"fs-id1167831871436\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f51cafd69c4fcea0787f414dc3d188c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#48;&#118;&#43;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"114\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831872096\">\n<div data-type=\"problem\" id=\"fs-id1167831872098\">\n<p id=\"fs-id1167831894505\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa633541797752340879cf43294f8b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#52;&#115;&#43;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834376359\">\n<p id=\"fs-id1167834357098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6aad726438cbffb425633cfe6b4d8c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#115;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"67\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831106542\">\n<div data-type=\"problem\" id=\"fs-id1167831106544\">\n<p id=\"fs-id1167831106546\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-648d2499b2f465e19e8f82e340659534_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#115;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#53;&#52;&#115;&#43;&#49;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"138\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832214525\">\n<div data-type=\"problem\" id=\"fs-id1167832214527\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3252f12f42d3765efb66fa10088b99a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#120;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831919669\">\n<p id=\"fs-id1167834426188\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02fe417d859369b58e21c83371ffb7f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#48;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448696\">\n<div data-type=\"problem\" id=\"fs-id1167834448698\">\n<p id=\"fs-id1167834448701\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b5b3fd784cbf7dcab6180b7746d50d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#122;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"113\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830700797\">\n<div data-type=\"problem\" id=\"fs-id1167830700799\">\n<p id=\"fs-id1167834185751\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f1e3e9238b83f833378f8f8156ea57b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#48;&#110;&#43;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"144\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834228726\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcfad050aef0e5cb54cf11ce68c8b2d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#110;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978453\">\n<div data-type=\"problem\" id=\"fs-id1167826978455\">\n<p id=\"fs-id1167826978457\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90a0d7cca00106eac3c0d411617f1ad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#50;&#112;&#43;&#49;&#54;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835419328\">\n<div data-type=\"problem\" id=\"fs-id1167835419330\">\n<p id=\"fs-id1167831887850\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa41baa7285626bbea41cceea33e3737_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#56;&#120;&#121;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167828421298\">\n<p id=\"fs-id1167828421300\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0baef7038d564bd17f3953ba7ef90b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#120;&#43;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834429403\">\n<div data-type=\"problem\" id=\"fs-id1167835328832\">\n<p id=\"fs-id1167835328834\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74a1eb2efd3d3ee935fdad18f8301538_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#48;&#114;&#115;&#43;&#51;&#54;&#123;&#115;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"145\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835351862\">\n<p id=\"fs-id1167835351865\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2cff3e839627f8d80309cb3290b54364_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#48;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#50;&#121;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831895344\">\n<p id=\"fs-id1167831895346\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00d41b01783140b999a12443e2a25253_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#48;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830865346\">\n<div data-type=\"problem\" id=\"fs-id1167830865348\">\n<p id=\"fs-id1167830865351\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8044d62d4751c3cce5592922e442cda8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#52;&#109;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834120730\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834120734\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7c62c7409959ca80be1a9824d8c696e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#106;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#43;&#56;&#48;&#106;&#107;&#43;&#49;&#54;&#48;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834396515\">\n<p id=\"fs-id1167834419478\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fd43eb16f429f7ed60460176f31abbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#106;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"86\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834179803\">\n<div data-type=\"problem\" id=\"fs-id1167834179805\">\n<p id=\"fs-id1167834179807\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0aa27179ba1e53360702407e4f475710_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;&#45;&#57;&#54;&#120;&#121;&#43;&#51;&#54;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"152\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834131742\">\n<div data-type=\"problem\" id=\"fs-id1167834131744\">\n<p id=\"fs-id1167834131746\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da5f8545865a58669df140937a3c5c42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#53;&#123;&#117;&#125;&#94;&#123;&#52;&#125;&#45;&#51;&#48;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#118;&#43;&#51;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#118;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"167\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835512152\">\n<p id=\"fs-id1167835512154\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c0179b56932dc9b5462a4b1da59ab914_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#117;&#45;&#118;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835410277\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832211896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a27f80a79ef4359b157d87d46e4683b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#48;&#123;&#112;&#125;&#94;&#123;&#52;&#125;&#43;&#51;&#48;&#48;&#123;&#112;&#125;&#94;&#123;&#52;&#125;&#113;&#43;&#50;&#53;&#48;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167832006550\"><strong data-effect=\"bold\">Factor Differences of Squares<\/strong><\/p>\n<p id=\"fs-id1167832006555\">In the following exercises, factor completely using the difference of squares pattern, if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832006559\">\n<div data-type=\"problem\" id=\"fs-id1167832006561\">\n<p id=\"fs-id1167835328059\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ec0320d924e6b6fef9fe1901f3b57df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835514103\">\n<p id=\"fs-id1167835514105\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f386dbdf9b23cc41aabc25e700e8785_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#118;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#118;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832042149\">\n<div data-type=\"problem\" id=\"fs-id1167832042151\">\n<p id=\"fs-id1167832042153\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3e07c84c69491127411fa51dce377e5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835489432\">\n<div data-type=\"problem\" id=\"fs-id1167835489434\">\n<p id=\"fs-id1167835489436\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d3aebdc89c8ab141320c9c6ae7b66cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#45;&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834533145\">\n<p id=\"fs-id1167834533147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a4b510faf2648c613764073077acedb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835258483\">\n<div data-type=\"problem\" id=\"fs-id1167835258485\">\n<p id=\"fs-id1167835258488\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00777b0e290bd6d1ff4d20276f469815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#45;&#50;&#53;&#123;&#115;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"80\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826996408\">\n<div data-type=\"problem\" id=\"fs-id1167826996410\">\n<p id=\"fs-id1167835629266\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b4f2ca0d1abfbb93e419b260b65ec31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832151088\">\n<p id=\"fs-id1167832151090\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-024dfe88f72cf7971bdf1abe5b64dd74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#113;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835420249\">\n<div data-type=\"problem\" id=\"fs-id1167835420251\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02319ee6457b0a0077bcbab25ef4bc64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#56;&#123;&#114;&#125;&#94;&#123;&#51;&#125;&#45;&#55;&#50;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"82\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832059445\">\n<div data-type=\"problem\" id=\"fs-id1167832059447\">\n<p id=\"fs-id1167832059449\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ad9113284fb05fc84c3ab7fcb66c75b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830865848\">\n<p id=\"fs-id1167834489216\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b826d5d64b9b53ac3db3be04831e3c96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834472427\">\n<div data-type=\"problem\" id=\"fs-id1167834397195\">\n<p id=\"fs-id1167834397197\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d46a3d7bbfe89912974de17b4d1d6b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834430179\">\n<div data-type=\"problem\" id=\"fs-id1167834430182\">\n<p id=\"fs-id1167834430184\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7462db34e3042010d36c1fda8b8f2ea1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831846725\">\n<p id=\"fs-id1167831846727\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e9ebd7ed4ac0500da34e229c76eeb9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#120;&#45;&#49;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#120;&#43;&#49;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"182\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063544\">\n<div data-type=\"problem\" id=\"fs-id1167834063546\">\n<p id=\"fs-id1167834063548\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f2fcae1beef0d8f1c2aa7bcae323026_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834382416\">\n<div data-type=\"problem\" id=\"fs-id1167834382418\">\n<p id=\"fs-id1167834382420\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-802d76a7548c303a7bc3399ebb67b944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#57;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#54;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827943898\">\n<p id=\"fs-id1167827943901\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14ff7d49159a2f6fee35baf610c15800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#99;&#45;&#54;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#51;&#99;&#43;&#54;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"159\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511779\">\n<div data-type=\"problem\" id=\"fs-id1167835511781\">\n<p id=\"fs-id1167835511783\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eea4d75f654c9760061434c88649db92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#57;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830964523\">\n<div data-type=\"problem\" id=\"fs-id1167834387520\">\n<p id=\"fs-id1167834387522\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b441a1b616d40218a901d4a516457cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#122;&#125;&#94;&#123;&#52;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834464139\">\n<p id=\"fs-id1167834464141\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4006d97849bade2da1461a808a65c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#122;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#122;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"200\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826870097\">\n<div data-type=\"problem\" id=\"fs-id1167826870099\">\n<p id=\"fs-id1167826870101\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-13df1e2d6644da7fdcaafea0ade38505_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#45;&#123;&#110;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834066085\">\n<div data-type=\"problem\" id=\"fs-id1167834066087\">\n<p id=\"fs-id1167834066089\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fe3d305ae39c013daa4bcd395af7a20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#50;&#123;&#97;&#125;&#94;&#123;&#52;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"113\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826779061\">\n<p id=\"fs-id1167826779063\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-643f32499fd91cda21d72c825871e41e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#97;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"229\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832057284\">\n<div data-type=\"problem\" id=\"fs-id1167831112600\">\n<p id=\"fs-id1167831112602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69908679fa5cae3cef5b54b4cba9df3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#109;&#125;&#94;&#123;&#52;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#51;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832220574\">\n<div data-type=\"problem\" id=\"fs-id1167832220576\">\n<p id=\"fs-id1167832220578\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b146ad6500c7a8950908804a21113775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#120;&#43;&#54;&#52;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835346291\">\n<p id=\"fs-id1167835346293\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-552a9d521b6ea02ad0ab1701bf63382d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#45;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#56;&#43;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831852240\">\n<div data-type=\"problem\" id=\"fs-id1167831852243\">\n<p id=\"fs-id1167831852245\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aade70f192e56743b6934e38b76ba2c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#52;&#112;&#43;&#52;&#57;&#45;&#123;&#113;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835518428\">\n<div data-type=\"problem\" id=\"fs-id1167835518430\">\n<p id=\"fs-id1167835518432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32f18cd39ab3d3fc26c8492f806641bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#97;&#43;&#57;&#45;&#57;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831040672\">\n<p id=\"fs-id1167831040675\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26d27d53b8dc797c4b58bf2fbc57b0a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#51;&#45;&#51;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#51;&#43;&#51;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"185\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167832058576\">\n<p id=\"fs-id1167832058578\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bd35d475004a75b83dd2b98b67f7326_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#109;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#109;&#43;&#57;&#45;&#49;&#54;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"158\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835531568\"><strong data-effect=\"bold\">Factor Sums and Differences of Cubes<\/strong><\/p>\n<p id=\"fs-id1167835531574\">In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835531578\">\n<div data-type=\"problem\" id=\"fs-id1167835531580\">\n<p id=\"fs-id1167835531582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-505499bdaab8fc45d31d897055ee8f1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"65\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831892971\">\n<p id=\"fs-id1167831892973\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44398c41da41c70087dc98990b5fc0b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#53;&#120;&#43;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"169\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834059830\">\n<div data-type=\"problem\" id=\"fs-id1167834059833\">\n<p id=\"fs-id1167834059835\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20703d25a32249f07c0bc0dea85975f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#54;&#125;&#43;&#53;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826873686\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834284123\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2a3839c2504d927663117af74a42adb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#122;&#125;&#94;&#123;&#54;&#125;&#45;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834284138\">\n<p id=\"fs-id1167831112415\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-614445be4e8dc7b4ea2e413fac2fe3f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#122;&#125;&#94;&#123;&#52;&#125;&#43;&#51;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"173\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826781270\">\n<div data-type=\"problem\" id=\"fs-id1167826781272\">\n<p id=\"fs-id1167834228054\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-962e1602968424acc8c1eb33c5734a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978324\">\n<div data-type=\"problem\" id=\"fs-id1167826978326\">\n<p id=\"fs-id1167826978328\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2990f5c5aa5e72502f9ffbdf95502e19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#45;&#51;&#52;&#51;&#123;&#116;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830702923\">\n<p id=\"fs-id1167830702925\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d2cc60bd229f3c74f294c5cb6e94719_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#45;&#55;&#116;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#43;&#49;&#52;&#116;&#43;&#52;&#57;&#123;&#116;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"184\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831894684\">\n<div data-type=\"problem\" id=\"fs-id1167831894686\">\n<p id=\"fs-id1167831894688\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97df07336c45edf58fcf40cb8a86a682_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#53;&#45;&#50;&#55;&#123;&#119;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835216068\">\n<div data-type=\"problem\" id=\"fs-id1167835216070\">\n<p id=\"fs-id1167835216072\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fda363dc8a2b262dd88c156bcf5bd134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#45;&#49;&#50;&#53;&#123;&#122;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835530885\">\n<p id=\"fs-id1167835530887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d9cee7a0fbb1f7422d34a0887e5a4d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#121;&#45;&#53;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#48;&#121;&#122;&#43;&#50;&#53;&#123;&#122;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"227\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828447087\">\n<div data-type=\"problem\" id=\"fs-id1167828447089\">\n<p id=\"fs-id1167828447091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d4a508a5d0353300faf6277a40797e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831832556\">\n<div data-type=\"problem\" id=\"fs-id1167832092252\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1dfa056951a00306d8b749e8e1b47150_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#49;&#54;&#123;&#97;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#50;&#53;&#123;&#98;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832092276\">\n<p id=\"fs-id1167832092278\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bd36f8fe5deb104380b5b445d26e7e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#97;&#43;&#53;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#54;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#97;&#98;&#43;&#50;&#53;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"232\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834098812\">\n<div data-type=\"problem\" id=\"fs-id1167834111477\">\n<p id=\"fs-id1167834111479\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-398b3cf71e21002ea2f57df114692e37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#121;&#125;&#94;&#123;&#51;&#125;&#43;&#56;&#123;&#122;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834562510\">\n<div data-type=\"problem\" id=\"fs-id1167835410116\">\n<p id=\"fs-id1167835410118\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59928e1927b266ef37995b5f220888f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#123;&#107;&#125;&#94;&#123;&#51;&#125;&#43;&#53;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835410135\">\n<p id=\"fs-id1167835410137\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3043eba14634e9ec0ce43b19bfbf64d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#107;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#107;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"171\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834556928\">\n<div data-type=\"problem\" id=\"fs-id1167834556931\">\n<p id=\"fs-id1167834556933\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68dc289b4a44a67892715f0f6360050c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#45;&#52;&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831883770\">\n<div data-type=\"problem\" id=\"fs-id1167831883772\">\n<p id=\"fs-id1167831883774\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb4fd67f92e19634ffabc35e3ab7395d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"100\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835517793\">\n<p id=\"fs-id1167835517795\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d72b7c73dc230747fa0d986b7e112b4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#50;&#121;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"206\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835325286\">\n<div data-type=\"problem\" id=\"fs-id1167835325288\">\n<p id=\"fs-id1167835325290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f9bcd793c2f69d0c8bc4e86e467489e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#51;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834373428\">\n<div data-type=\"problem\" id=\"fs-id1167834373430\">\n<p id=\"fs-id1167834373432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05b61bdbffd847a55ee18f3400de1ff8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#43;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834289482\">\n<p id=\"fs-id1167834289484\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-753e9905d364a685015188ec123c2c0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"132\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831071390\">\n<div data-type=\"problem\" id=\"fs-id1167831071392\">\n<p id=\"fs-id1167831071394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f3a098e201c5877a827bf951b97acb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#55;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"118\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535070\">\n<div data-type=\"problem\" id=\"fs-id1167834535072\">\n<p id=\"fs-id1167834535074\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2bef9e2ad9db588f596ab9c0d242e48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835166662\">\n<p id=\"fs-id1167835166664\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c3069714d241ace635333b41a3d340e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#121;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#49;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#121;&#43;&#50;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"202\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067655\">\n<div data-type=\"problem\" id=\"fs-id1167832067657\">\n<p id=\"fs-id1167832067659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ecedfdb7714da9d33702aaf582f235f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831116304\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p>\n<p id=\"fs-id1167831116311\">In the following exercises, factor completely.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831116314\">\n<div data-type=\"problem\" id=\"fs-id1167831116316\">\n<p id=\"fs-id1167831116318\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7b08b4c6bfbbb802b6a800eab78cedf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#52;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835390217\">\n<p id=\"fs-id1167835390220\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc17c92d7b730b0b33543604bb6036d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#97;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#97;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831958100\">\n<div data-type=\"problem\" id=\"fs-id1167831958102\">\n<p id=\"fs-id1167831958104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ee9f36db03630aadb0ca5076530f2dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835509468\">\n<div data-type=\"problem\" id=\"fs-id1167835509470\">\n<p id=\"fs-id1167835509472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-115d104409e9cc347dcf69ff6ea5970d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835509489\">\n<p id=\"fs-id1167835509491\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c809a08c04bcff04343d1d94caa0e93b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#113;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#113;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835357654\">\n<div data-type=\"problem\" id=\"fs-id1167835357656\">\n<p id=\"fs-id1167835357659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4aae66c96c86995022ab66bcd70d72d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831921204\">\n<div data-type=\"problem\" id=\"fs-id1167831921206\">\n<p id=\"fs-id1167831921208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5138dafcae436118189c1e9998ceee1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#55;&#50;&#120;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"123\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831921231\">\n<p id=\"fs-id1167831948996\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0551c30cab6fcc010b356c7c207e6c8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#120;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"69\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831949022\">\n<div data-type=\"problem\" id=\"fs-id1167831949024\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c014ce02b87e306e13260db9065273b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#121;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"113\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621469\">\n<div data-type=\"problem\" id=\"fs-id1167835621471\">\n<p id=\"fs-id1167835621474\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbdec160cd44a6f3597d8b136c29f61f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167830964433\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39d60e2819ef176b8eedf9e31fce4cb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#123;&#112;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826873436\">\n<div data-type=\"problem\" id=\"fs-id1167826873438\">\n<p id=\"fs-id1167826873441\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7def13f9664ddb39e8666f6ac7be5fae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#49;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#54;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835510252\">\n<div data-type=\"problem\" id=\"fs-id1167835510255\">\n<p id=\"fs-id1167835510257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b1fe283686926cb233ca53d7491f9ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#53;&#45;&#56;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835510274\">\n<p id=\"fs-id1167835510276\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dd212a1fd2f3f6e437af3e99440bbae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#45;&#50;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#53;&#43;&#49;&#48;&#121;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"193\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826864191\">\n<div data-type=\"problem\" id=\"fs-id1167826864193\">\n<p id=\"fs-id1167826864196\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ceaaa6c8fd8a021bcd5f309b73b1f29a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#55;&#123;&#117;&#125;&#94;&#123;&#51;&#125;&#43;&#49;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"93\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830700659\">\n<div data-type=\"problem\" id=\"fs-id1167830700661\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e88c2842eea925fb4f8de18f1df81128_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#53;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#48;&#110;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"126\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832066083\">\n<p id=\"fs-id1167832066086\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9344792f65bf0f323eceb14501c3ed4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826873718\">\n<div data-type=\"problem\" id=\"fs-id1167826873720\">\n<p id=\"fs-id1167826873723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f872c9de286243208ea6a0b03d8cb122_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#56;&#123;&#113;&#125;&#94;&#123;&#51;&#125;&#45;&#50;&#52;&#123;&#113;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826799330\">\n<div data-type=\"problem\" id=\"fs-id1167826799332\">\n<p id=\"fs-id1167826799335\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5618dd45b52f4e39da712c95daf20d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#48;&#120;&#43;&#50;&#53;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835513292\">\n<p id=\"fs-id1167835513295\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cae559509ef1ad4dbb1d3df23f52664d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"172\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835319121\">\n<div data-type=\"problem\" id=\"fs-id1167835319123\">\n<p id=\"fs-id1167835319126\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6be679396c4ecff5b8e2997d8834dc97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#120;&#43;&#51;&#54;&#45;&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"145\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832152607\">\n<div data-type=\"problem\" id=\"fs-id1167832152609\">\n<p id=\"fs-id1167832152611\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65837c9d0361fb0663f0648c7d97cafe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#43;&#56;&#123;&#120;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835357746\">\n<p id=\"fs-id1167835357748\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3510018e7392ef3752e8c1808304a181_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"137\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167828426665\">\n<div data-type=\"problem\" id=\"fs-id1167828426667\">\n<p id=\"fs-id1167828426669\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1cd9717624759162bd69c6279d309903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#45;&#54;&#52;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167831872220\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167834394383\">\n<div data-type=\"problem\" id=\"fs-id1167834394385\">\n<p id=\"fs-id1167834394387\">Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834394393\">\n<p>Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834394400\">\n<div data-type=\"problem\" id=\"fs-id1167834394402\">\n<p id=\"fs-id1167834394404\">How do you recognize the binomial squares pattern?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831117390\">\n<div data-type=\"problem\" id=\"fs-id1167831117393\">\n<p id=\"fs-id1167831117395\">Explain why <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00e13a4423b08862373794e5ef50f37e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#92;&#110;&#101;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -4px;\" \/> Use algebra, words, or pictures.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834525248\">\n<p id=\"fs-id1167834525250\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834525255\">\n<div data-type=\"problem\" id=\"fs-id1167834525257\">\n<p id=\"fs-id1167834525259\">Maribel factored <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08e8880012ce9d39f44f9d15b0298185_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#51;&#48;&#121;&#43;&#56;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/> as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-78c8b122f552b27d264ebfb93e68a3fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"64\" style=\"vertical-align: -4px;\" \/> Was she right or wrong? How do you know?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167832151236\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167832151241\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167831919620\" data-alt=\"This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_06_03_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167831919631\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":103,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3102","chapter","type-chapter","status-publish","hentry"],"part":2962,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3102","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3102\/revisions"}],"predecessor-version":[{"id":15245,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3102\/revisions\/15245"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2962"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3102\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3102"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3102"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3102"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3102"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}