{"id":602,"date":"2019-04-29T15:59:07","date_gmt":"2019-04-29T19:59:07","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=602"},"modified":"2020-02-03T14:02:59","modified_gmt":"2020-02-03T19:02:59","slug":"7-5-factoring-special-products","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/7-5-factoring-special-products\/","title":{"raw":"7.5 Factoring Special Products","rendered":"7.5 Factoring Special Products"},"content":{"raw":"[latexpage]\r\n\r\nNow transition from multiplying special products to factoring special products. If you can recognize them, you can save a lot of time. The following is a list of these special products (note that a2 <\/sup>+ b2<\/sup> cannot be factored):\r\n

\\(\\begin{array}{lll}\r\na^2-b^2&=&(a+b)(a-b) \\\\\r\n(a+b)^2&=&a^2+2ab+b^2 \\\\\r\n(a-b)^2&=&a^2-2ab+b^2 \\\\\r\na^3-b^3&=&(a-b)(a^2+ab+b^2) \\\\\r\na^3+b^3&=&(a+b)(a^2-ab+b^2) \\\\\r\n\\end{array}\\)<\/p>\r\nThe challenge is therefore in recognizing the special product.\r\n

\r\n

Example 7.5.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor \\(x^2 - 36\\).\r\n\r\nThis is a difference of squares. \\((x - 6)(x + 6)\\) is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor \\(x^2 - 6x + 9\\).\r\n\r\nThis is a perfect square. \\((x - 3)(x - 3)\\) or \\((x - 3)^2\\) is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor \\(x^2 + 6x + 9\\).\r\n\r\nThis is a perfect square. \\((x + 3)(x + 3)\\) or \\((x + 3)^2\\) is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor \\(4x^2 + 20xy + 25y^2\\).\r\n\r\nThis is a perfect square. \\((2x + 5y)(2x + 5y)\\) or \\((2x + 5y)^2\\) is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.5<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor \\(m^3 - 27\\).\r\n\r\nThis is a difference of cubes. \\((m - 3)(m^2 + 3m + 9)\\) is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n
\r\n

Example 7.5.6<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFactor \\(125p^3 + 8r^3\\).\r\n\r\nThis is a difference of cubes. \\((5p + 2r)(25p^2 - 10pr + 4r^2)\\) is the solution.\r\n\r\n<\/div>\r\n<\/div>\r\n

Questions<\/h1>\r\nFactor each of the following polynomials.\r\n
    \r\n \t
  1. \\(r^2-16\\)<\/li>\r\n \t
  2. \\(x^2-9\\)<\/li>\r\n \t
  3. \\(v^2-25\\)<\/li>\r\n \t
  4. \\(x^2-1\\)<\/li>\r\n \t
  5. \\(p^2-4\\)<\/li>\r\n \t
  6. \\(4v^2-1\\)<\/li>\r\n \t
  7. \\(3x^2-27\\)<\/li>\r\n \t
  8. \\(5n^2-20\\)<\/li>\r\n \t
  9. \\(16x^2-36\\)<\/li>\r\n \t
  10. \\(125x^2+45y^2\\)<\/li>\r\n \t
  11. \\(a^2-2a+1\\)<\/li>\r\n \t
  12. \\(k^2+4k+4\\)<\/li>\r\n \t
  13. \\(x^2+6x+9\\)<\/li>\r\n \t
  14. \\(n^2-8n+16\\)<\/li>\r\n \t
  15. \\(25p^2-10p+1\\)<\/li>\r\n \t
  16. \\(x^2+2x+1\\)<\/li>\r\n \t
  17. \\(25a^2+30ab+9b^2\\)<\/li>\r\n \t
  18. \\(x^2+8xy+16y^2\\)<\/li>\r\n \t
  19. \\(8x^2-24xy+18y^2\\)<\/li>\r\n \t
  20. \\(20x^2+20xy+5y^2\\)<\/li>\r\n \t
  21. \\(8-m^3\\)<\/li>\r\n \t
  22. \\(x^3+64\\)<\/li>\r\n \t
  23. \\(x^3-64\\)<\/li>\r\n \t
  24. \\(x^3+8\\)<\/li>\r\n \t
  25. \\(216-u^3\\)<\/li>\r\n \t
  26. \\(125x^3-216\\)<\/li>\r\n \t
  27. \\(125a^3-64\\)<\/li>\r\n \t
  28. \\(64x^3-27\\)<\/li>\r\n \t
  29. \\(64x^3+27y^3\\)<\/li>\r\n \t
  30. \\(32m^3-108n^3\\)<\/li>\r\n<\/ol>\r\nAnswer Key 7.5<\/a>","rendered":"

    Now transition from multiplying special products to factoring special products. If you can recognize them, you can save a lot of time. The following is a list of these special products (note that a2 <\/sup>+ b2<\/sup> cannot be factored):<\/p>\n

    \"\begin{array}{lll}<\/p>\n

    The challenge is therefore in recognizing the special product.<\/p>\n

    \n
    \n

    Example 7.5.1<\/p>\n<\/header>\n

    \n

    Factor \"x^2 - 36\".<\/p>\n

    This is a difference of squares. \"(x - 6)(x + 6)\" is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.2<\/p>\n<\/header>\n

    \n

    Factor \"x^2 - 6x + 9\".<\/p>\n

    This is a perfect square. \"(x - 3)(x - 3)\" or \"(x - 3)^2\" is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.3<\/p>\n<\/header>\n

    \n

    Factor \"x^2 + 6x + 9\".<\/p>\n

    This is a perfect square. \"(x + 3)(x + 3)\" or \"(x + 3)^2\" is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.4<\/p>\n<\/header>\n

    \n

    Factor \"4x^2 + 20xy + 25y^2\".<\/p>\n

    This is a perfect square. \"(2x + 5y)(2x + 5y)\" or \"(2x + 5y)^2\" is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.5<\/p>\n<\/header>\n

    \n

    Factor \"m^3 - 27\".<\/p>\n

    This is a difference of cubes. \"(m - 3)(m^2 + 3m + 9)\" is the solution.<\/p>\n<\/div>\n<\/div>\n

    \n
    \n

    Example 7.5.6<\/p>\n<\/header>\n

    \n

    Factor \"125p^3 + 8r^3\".<\/p>\n

    This is a difference of cubes. \"(5p + 2r)(25p^2 - 10pr + 4r^2)\" is the solution.<\/p>\n<\/div>\n<\/div>\n

    Questions<\/h1>\n

    Factor each of the following polynomials.<\/p>\n

      \n
    1. \"r^2-16\"<\/li>\n
    2. \"x^2-9\"<\/li>\n
    3. \"v^2-25\"<\/li>\n
    4. \"x^2-1\"<\/li>\n
    5. \"p^2-4\"<\/li>\n
    6. \"4v^2-1\"<\/li>\n
    7. \"3x^2-27\"<\/li>\n
    8. \"5n^2-20\"<\/li>\n
    9. \"16x^2-36\"<\/li>\n
    10. \"125x^2+45y^2\"<\/li>\n
    11. \"a^2-2a+1\"<\/li>\n
    12. \"k^2+4k+4\"<\/li>\n
    13. \"x^2+6x+9\"<\/li>\n
    14. \"n^2-8n+16\"<\/li>\n
    15. \"25p^2-10p+1\"<\/li>\n
    16. \"x^2+2x+1\"<\/li>\n
    17. \"25a^2+30ab+9b^2\"<\/li>\n
    18. \"x^2+8xy+16y^2\"<\/li>\n
    19. \"8x^2-24xy+18y^2\"<\/li>\n
    20. \"20x^2+20xy+5y^2\"<\/li>\n
    21. \"8-m^3\"<\/li>\n
    22. \"x^3+64\"<\/li>\n
    23. \"x^3-64\"<\/li>\n
    24. \"x^3+8\"<\/li>\n
    25. \"216-u^3\"<\/li>\n
    26. \"125x^3-216\"<\/li>\n
    27. \"125a^3-64\"<\/li>\n
    28. \"64x^3-27\"<\/li>\n
    29. \"64x^3+27y^3\"<\/li>\n
    30. \"32m^3-108n^3\"<\/li>\n<\/ol>\n

      Answer Key 7.5<\/a><\/p>\n","protected":false},"author":540,"menu_order":19,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-602","chapter","type-chapter","status-publish","hentry"],"part":379,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/602","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/540"}],"version-history":[{"count":9,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/602\/revisions"}],"predecessor-version":[{"id":3792,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/602\/revisions\/3792"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/parts\/379"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/602\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=602"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=602"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=602"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}