{"id":561,"date":"2019-04-29T14:45:34","date_gmt":"2019-04-29T18:45:34","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=chapter&p=561"},"modified":"2019-12-03T16:43:44","modified_gmt":"2019-12-03T21:43:44","slug":"6-4-basic-operations-using-polynomials","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/chapter\/6-4-basic-operations-using-polynomials\/","title":{"raw":"6.4 Basic Operations Using Polynomials","rendered":"6.4 Basic Operations Using Polynomials"},"content":{"raw":"[latexpage]\r\n\r\nMany applications in mathematics have to do with what are called polynomials. Polynomials are made up of terms. Terms are a product of numbers and\/or variables. For example, \\(5x\\), \\(2y^2\\), \\(-5\\), \\(ab^3c\\), and \\(x\\) are all terms. Terms are connected to each other by addition or subtraction.\r\n\r\nExpressions are often defined by the number of terms they have.\r\n

\\(\\begin{array}{l}\r\n\\text{A monomial has one term, such as } x, xy, 3x^2.\\\\\r\n\\text{A binomial has two terms connected by a + or -, such as } a^2-b^2, 3x-y, 4x+2xy^3 \\\\\r\n\\text{A trinomial has three terms connected by a + or -, such as } ax^2 + bx + c\r\n\\end{array}\\)<\/p>\r\nThe term polynomial is generic for many terms. Monomials, binomials, trinomials, and expressions with more terms all fall under the umbrella of \u201cpolynomials.\u201d\r\n\r\nPolynomials are classified by the sum of exponents of the term with the highest exponent sum, which is called the degree of the polynomial.\r\n\r\n \r\n

A first degree polynomial<\/strong> is a linear polynomial and would not have any terms with a sum of exponents greater than one. Examples include \\(3x + 2y + 4z\\), \\(42x - 56y\\), \\(22z\\).<\/p>\r\n

A second degree polynomial<\/strong> is a quadratic polynomial and\u00a0would not have any terms with a sum of exponents greater than two. Examples include \\(3x^2 + 2x + 4y^2\\), \\(42xy - 3z\\), \\(2zx\\).<\/p>\r\n

A third degree polynomial<\/strong> is a cubic polynomial and\u00a0would not have any terms with a sum of exponents greater than three. Examples include \\(5x^2 + 2xy^2 + 4y^2\\), \\(42xyz - 5z^2\\), \\(3zx^2\\).<\/p>\r\n

A fourth degree polynomial<\/strong> is a quartic polynomial and would not have any terms with a sum of exponents greater than four. Examples include \\(5x^2 + 3xy^2z + 4y^2\\), \\(42xyz - 6z^4\\), \\(8z^3x\\).<\/p>\r\n

A fifth degree polynomial<\/strong> is a quintic polynomial.<\/p>\r\n

A sixth degree polynomial<\/strong> is a sextic polynomial.<\/p>\r\n

A seventh degree polynomial<\/strong> is a septic polynomial.<\/p>\r\n

A eighth degree polynomial<\/strong> is a octic polynomial.<\/p>\r\n

A ninth degree polynomial<\/strong> is a nonic polynomial.<\/p>\r\n

A tenth degree polynomial<\/strong> is a decic polynomial.<\/p>\r\n \r\n\r\nThe degree of any term is the sum of its exponents:\r\n

\\(x^9, x^7y^2, x^3y^3z^3\\) are all ninth degree terms (nonic polynomial)<\/p>\r\n

\\(x^6, x^4y^2, x^3yz^2\\) are all sixth degree terms (sextic polynomial)<\/p>\r\n

\\(x^4, x^2y^2, xyz^2\\) are all fourth degree terms (quartic polynomial)<\/p>\r\nTerms of a polynomial are named in the order of their appearance. For instance, the polynomial \\(2x^4y + x^3y^2 - x^2y^3 + 5xy^4\\) has four terms, each one of the fifth degree. The terms are numbered in order for this polynomial, starting from the first term \\((2x^4y)\\) and continuing to the second ((x^3y^2)\\), third \\((-x^2y^3)\\), and fourth terms \\((5xy^4)\\).\r\n\r\nIf it is known what the variable in a polynomial represents, it is possible to substitute in the value and evaluate the polynomial, as shown in the following example.\r\n

\r\n

Example 6.4.1<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify the expression \\(2x^2 - 4x + 6\\) using \\(x = -4\\).\r\n

\\(\\begin{array}{rlllll}\r\n\\text{When } x=-4, \\text{ replace all }x\\text{ with }-4:&2x^2&-&4x&+&6 \\\\\r\n\\text{Reduce the exponents:}&2(-4)^2&-&4(-4)&+&6 \\\\\r\n\\text{Multiply all coefficients:}&2(16)&-&4(-4)&+&6 \\\\\r\n\\text{Add:}&32&+&16&+&6 \\\\\r\n\\text{Solution:}&54&&&&\r\n\\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nRemember the exponent only affects the number to which it is physically attached. This means \u221232<\/sup> = \u22129 because the exponent is only attached to the 3, whereas (\u22123)2<\/sup> = 9 because the exponent is attached to the parentheses and everything inside.\r\n

\r\n

Example 6.4.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify the expression \\(-x^2 + 2x + 6\\) using \\(x = 3\\).\r\n

\\(\\begin{array}{rlllll}\r\n\\text{When }x=3,\\text{ replace all }x \\text{ with 3:}&-x^2&+&2x&+&6 \\\\\r\n\\text{Reduce the exponents:}&-(3)^2&+&2(3)&+&6 \\\\\r\n\\text{Multiply:}&-9&+&2(3)&+&6 \\\\\r\n\\text{Add:}&-9&+&6&+&6 \\\\\r\n\\text{Solution:}&3&&&&\r\n\\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Sometimes, when working with polynomials,\u00a0 the value of the variable is unknown and the polynomial will be simplified rather than ending with some value. The simplest operation with polynomials is addition. When adding polynomials, it is merely combining like terms. Consider the following example.<\/span><\/p>\r\n\r\n

\r\n

Example 6.4.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify the expression \\((4x^3 - 2x + 8) + (3x^3 - 9x^2 - 11).\\)\r\n\r\nThe first thing one should do is place the equations one over top of the other, ordering each of the terms into columns so they can simply be added or subtracted.\r\n

\\(\\begin{array}{rrrrrrrl}\r\n4x^3&&&-&2x&+&8& \\\\\r\n3x^3&-&9x^2&&&-&11&\\text{Add each of the columns} \\\\\r\n\\midrule\r\n7x^3&-&9x^2&-&2x&-&3&\\text{Solution} \\\\\r\n\\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Subtracting polynomials is almost as fast as addition. Subtraction is generally shown by a minus sign in front of the parentheses. When there is a negative sign in front of parentheses,\u00a0 distribute it throughout the expression, changing the signs of everything inside. <\/span><\/p>\r\n\r\n

\r\n

Example 6.4.4<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify the expression \\((5x^2 - 2x + 7) - (3x^2 + 6x - 4).\\)\r\n

\\(\\begin{array}{rrrrrrl}\r\n&5x^2&-&2x&+&7& \\\\\r\n-&(3x^2&+&6x&-&4)&\\text{Subtract each bottom term from each top term} \\\\\r\n\\midrule\r\n&2x^2&-&8x&+&11&\\text{Solution}\r\n\\end{array}\\)<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n

Example 6.4.5<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nSimplify the expression \\((2x^2 - 4x + 3) + (5x^2 - 6x + 1) - (x^2 - 9x + 8).\\)\r\n\r\n\\(\\begin{array}{rrrrrrl}\r\n&2x^2&-&4x&+&3& \\\\\r\n&5x^2&-&6x&+&1& \\\\\r\n-&(x^2&-&9x&+&8)&\\text{Add and subtract} \\\\\r\n\\midrule\r\n&6x^2&-&x&-&4&\\text{Solution} \\\\\r\n\\end{array}\\)\r\n\r\n<\/div>\r\n<\/div>\r\n

Questions<\/h1>\r\nFor questions 1 to 8, simplify each expression using the variables given.\r\n
    \r\n \t
  1. \\(-a^3 - a^2 + 6a - 21 \\text{ when }a = -4\\)<\/li>\r\n \t
  2. \\(n^2 + 3n - 11 \\text{ when }n = -6\\)<\/li>\r\n \t
  3. \\(-5n^4 - 11n^3 - 9n^2 - n - 5 \\text{ when } n = -1\\)<\/li>\r\n \t
  4. \\(x^4 - 5x^3 - x + 13 \\text{ when } x = 5\\)<\/li>\r\n \t
  5. \\(x^2 + 9x + 23 \\text{ when } x = -3\\)<\/li>\r\n \t
  6. \\(-6x^3 + 41x^2 - 32x + 11 \\text{ when } x = 6\\)<\/li>\r\n \t
  7. \\(x^4 - 6x^3 + x^2 - 24 \\text{ when } x = 6\\)<\/li>\r\n \t
  8. \\(m^4 + 8m^3 + 14m^2 + 13m + 5 \\text{ when } m = -6\\)<\/li>\r\n<\/ol>\r\nFor questions 9 to 28, simplify the following expressions.\r\n
      \r\n \t
    1. \\((5p - 5p^4) - (8p - 8p^4)\\)<\/li>\r\n \t
    2. \\((7m^2 + 5m^3) - (6m^3 - 5m^2)\\)<\/li>\r\n \t
    3. \\((1 + 5p^3) - (1 - 8p^3)\\)<\/li>\r\n \t
    4. \\((6x^3 + 5x) - (8x + 6x^3)\\)<\/li>\r\n \t
    5. \\((5n^4 + 6n^3) + (8 - 3n^3 - 5n^4)\\)<\/li>\r\n \t
    6. \\((8x^2 + 1) - (6 - x^2 - x^4)\\)<\/li>\r\n \t
    7. \\((2a + 2a^4) - (3a^2 - 5a^4 + 4a)\\)<\/li>\r\n \t
    8. \\((6v + 8v^3) + (3 + 4v^3 - 3v)\\)<\/li>\r\n \t
    9. \\((4p^2 - 3 - 2p) - (3p^2 - 6p + 3)\\)<\/li>\r\n \t
    10. \\((7 + 4m + 8m^4) - (5m^4 + 1 + 6m)\\)<\/li>\r\n \t
    11. \\((3 + 2n^2 + 4n^4) + (n^3 - 7n^2 - 4n^4)\\)<\/li>\r\n \t
    12. \\((7x^2 + 2x^4 + 7x^3) + (6x^3 - 8x^4 - 7x^2)\\)<\/li>\r\n \t
    13. \\((8r^4 - 5r^3 + 5r^2) + (2r^2 + 2r^3 - 7r^4 + 1)\\)<\/li>\r\n \t
    14. \\((4x^3 + x - 7x^2) + (x^2 - 8 + 2x + 6x^3)\\)<\/li>\r\n \t
    15. \\((2n^2 + 7n^4 - 2) + (2 + 2n^3 + 4n^2 + 2n^4)\\)<\/li>\r\n \t
    16. \\((7b^3 - 4b + 4b^4) - (8b^3 - 4b^2 + 2b^4 - 8b)\\)<\/li>\r\n \t
    17. \\((8 - b + 7b^3) - (3b^4 + 7b - 8 + 7b^2) + (3 - 3b + 6b^3)\\)<\/li>\r\n \t
    18. \\((1 - 3n^4 - 8n^3) + (7n^4 + 2 - 6n^2 + 3n^3) + (4n^3 + 8n^4 + 7)\\)<\/li>\r\n \t
    19. \\((8x^4 + 2x^3 + 2x) + (2x + 2 - 2x^3 - x^4) - (x^3 + 5x^4 + 8x)\\)<\/li>\r\n \t
    20. \\((6x - 5x^4 - 4x^2) - (2x - 7x^2 - 4x^4 - 8) - (8 - 6x^2 - 4x^4)\\)<\/li>\r\n<\/ol>\r\nAnswer Key 6.4<\/a>","rendered":"

      Many applications in mathematics have to do with what are called polynomials. Polynomials are made up of terms. Terms are a product of numbers and\/or variables. For example, \"5x\", \"2y^2\", \"-5\", \"ab^3c\", and \"x\" are all terms. Terms are connected to each other by addition or subtraction.<\/p>\n

      Expressions are often defined by the number of terms they have.<\/p>\n

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

      The term polynomial is generic for many terms. Monomials, binomials, trinomials, and expressions with more terms all fall under the umbrella of \u201cpolynomials.\u201d<\/p>\n

      Polynomials are classified by the sum of exponents of the term with the highest exponent sum, which is called the degree of the polynomial.<\/p>\n

       <\/p>\n

      A first degree polynomial<\/strong> is a linear polynomial and would not have any terms with a sum of exponents greater than one. Examples include \"3x + 2y + 4z\", \"42x - 56y\", \"22z\".<\/p>\n

      A second degree polynomial<\/strong> is a quadratic polynomial and\u00a0would not have any terms with a sum of exponents greater than two. Examples include \"3x^2 + 2x + 4y^2\", \"42xy - 3z\", \"2zx\".<\/p>\n

      A third degree polynomial<\/strong> is a cubic polynomial and\u00a0would not have any terms with a sum of exponents greater than three. Examples include \"5x^2 + 2xy^2 + 4y^2\", \"42xyz - 5z^2\", \"3zx^2\".<\/p>\n

      A fourth degree polynomial<\/strong> is a quartic polynomial and would not have any terms with a sum of exponents greater than four. Examples include \"5x^2 + 3xy^2z + 4y^2\", \"42xyz - 6z^4\", \"8z^3x\".<\/p>\n

      A fifth degree polynomial<\/strong> is a quintic polynomial.<\/p>\n

      A sixth degree polynomial<\/strong> is a sextic polynomial.<\/p>\n

      A seventh degree polynomial<\/strong> is a septic polynomial.<\/p>\n

      A eighth degree polynomial<\/strong> is a octic polynomial.<\/p>\n

      A ninth degree polynomial<\/strong> is a nonic polynomial.<\/p>\n

      A tenth degree polynomial<\/strong> is a decic polynomial.<\/p>\n

       <\/p>\n

      The degree of any term is the sum of its exponents:<\/p>\n

      \"x^9, x^7y^2, x^3y^3z^3\" are all ninth degree terms (nonic polynomial)<\/p>\n

      \"x^6, x^4y^2, x^3yz^2\" are all sixth degree terms (sextic polynomial)<\/p>\n

      \"x^4, x^2y^2, xyz^2\" are all fourth degree terms (quartic polynomial)<\/p>\n

      Terms of a polynomial are named in the order of their appearance. For instance, the polynomial \"2x^4y + x^3y^2 - x^2y^3 + 5xy^4\" has four terms, each one of the fifth degree. The terms are numbered in order for this polynomial, starting from the first term \"(2x^4y)\" and continuing to the second ((x^3y^2)\\), third \"(-x^2y^3)\", and fourth terms \"(5xy^4)\".<\/p>\n

      If it is known what the variable in a polynomial represents, it is possible to substitute in the value and evaluate the polynomial, as shown in the following example.<\/p>\n

      \n
      \n

      Example 6.4.1<\/p>\n<\/header>\n

      \n

      Simplify the expression \"2x^2 - 4x + 6\" using \"x = -4\".<\/p>\n

      \"\begin{array}{rlllll}<\/p>\n<\/div>\n<\/div>\n

      Remember the exponent only affects the number to which it is physically attached. This means \u221232<\/sup> = \u22129 because the exponent is only attached to the 3, whereas (\u22123)2<\/sup> = 9 because the exponent is attached to the parentheses and everything inside.<\/p>\n

      \n
      \n

      Example 6.4.2<\/p>\n<\/header>\n

      \n

      Simplify the expression \"-x^2 + 2x + 6\" using \"x = 3\".<\/p>\n

      \"\begin{array}{rlllll}<\/p>\n<\/div>\n<\/div>\n

      Sometimes, when working with polynomials,\u00a0 the value of the variable is unknown and the polynomial will be simplified rather than ending with some value. The simplest operation with polynomials is addition. When adding polynomials, it is merely combining like terms. Consider the following example.<\/span><\/p>\n

      \n
      \n

      Example 6.4.3<\/p>\n<\/header>\n

      \n

      Simplify the expression \"(4x^3 - 2x + 8) + (3x^3 - 9x^2 - 11).\"<\/p>\n

      The first thing one should do is place the equations one over top of the other, ordering each of the terms into columns so they can simply be added or subtracted.<\/p>\n

      \"\begin{array}{rrrrrrrl}<\/p>\n<\/div>\n<\/div>\n

      Subtracting polynomials is almost as fast as addition. Subtraction is generally shown by a minus sign in front of the parentheses. When there is a negative sign in front of parentheses,\u00a0 distribute it throughout the expression, changing the signs of everything inside. <\/span><\/p>\n

      \n
      \n

      Example 6.4.4<\/p>\n<\/header>\n

      \n

      Simplify the expression \"(5x^2 - 2x + 7) - (3x^2 + 6x - 4).\"<\/p>\n

      \"\begin{array}{rrrrrrl}<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      Example 6.4.5<\/p>\n<\/header>\n

      \n

      Simplify the expression \"(2x^2 - 4x + 3) + (5x^2 - 6x + 1) - (x^2 - 9x + 8).\"<\/p>\n

      \"\begin{array}{rrrrrrl}<\/p>\n<\/div>\n<\/div>\n

      Questions<\/h1>\n

      For questions 1 to 8, simplify each expression using the variables given.<\/p>\n

        \n
      1. \"-a^3 - a^2 + 6a - 21 \text{ when }a = -4\"<\/li>\n
      2. \"n^2 + 3n - 11 \text{ when }n = -6\"<\/li>\n
      3. \"-5n^4 - 11n^3 - 9n^2 - n - 5 \text{ when } n = -1\"<\/li>\n
      4. \"x^4 - 5x^3 - x + 13 \text{ when } x = 5\"<\/li>\n
      5. \"x^2 + 9x + 23 \text{ when } x = -3\"<\/li>\n
      6. \"-6x^3 + 41x^2 - 32x + 11 \text{ when } x = 6\"<\/li>\n
      7. \"x^4 - 6x^3 + x^2 - 24 \text{ when } x = 6\"<\/li>\n
      8. \"m^4 + 8m^3 + 14m^2 + 13m + 5 \text{ when } m = -6\"<\/li>\n<\/ol>\n

        For questions 9 to 28, simplify the following expressions.<\/p>\n

          \n
        1. \"(5p - 5p^4) - (8p - 8p^4)\"<\/li>\n
        2. \"(7m^2 + 5m^3) - (6m^3 - 5m^2)\"<\/li>\n
        3. \"(1 + 5p^3) - (1 - 8p^3)\"<\/li>\n
        4. \"(6x^3 + 5x) - (8x + 6x^3)\"<\/li>\n
        5. \"(5n^4 + 6n^3) + (8 - 3n^3 - 5n^4)\"<\/li>\n
        6. \"(8x^2 + 1) - (6 - x^2 - x^4)\"<\/li>\n
        7. \"(2a + 2a^4) - (3a^2 - 5a^4 + 4a)\"<\/li>\n
        8. \"(6v + 8v^3) + (3 + 4v^3 - 3v)\"<\/li>\n
        9. \"(4p^2 - 3 - 2p) - (3p^2 - 6p + 3)\"<\/li>\n
        10. \"(7 + 4m + 8m^4) - (5m^4 + 1 + 6m)\"<\/li>\n
        11. \"(3 + 2n^2 + 4n^4) + (n^3 - 7n^2 - 4n^4)\"<\/li>\n
        12. \"(7x^2 + 2x^4 + 7x^3) + (6x^3 - 8x^4 - 7x^2)\"<\/li>\n
        13. \"(8r^4 - 5r^3 + 5r^2) + (2r^2 + 2r^3 - 7r^4 + 1)\"<\/li>\n
        14. \"(4x^3 + x - 7x^2) + (x^2 - 8 + 2x + 6x^3)\"<\/li>\n
        15. \"(2n^2 + 7n^4 - 2) + (2 + 2n^3 + 4n^2 + 2n^4)\"<\/li>\n
        16. \"(7b^3 - 4b + 4b^4) - (8b^3 - 4b^2 + 2b^4 - 8b)\"<\/li>\n
        17. \"(8 - b + 7b^3) - (3b^4 + 7b - 8 + 7b^2) + (3 - 3b + 6b^3)\"<\/li>\n
        18. \"(1 - 3n^4 - 8n^3) + (7n^4 + 2 - 6n^2 + 3n^3) + (4n^3 + 8n^4 + 7)\"<\/li>\n
        19. \"(8x^4 + 2x^3 + 2x) + (2x + 2 - 2x^3 - x^4) - (x^3 + 5x^4 + 8x)\"<\/li>\n
        20. \"(6x - 5x^4 - 4x^2) - (2x - 7x^2 - 4x^4 - 8) - (8 - 6x^2 - 4x^4)\"<\/li>\n<\/ol>\n

          Answer Key 6.4<\/a><\/p>\n","protected":false},"author":540,"menu_order":17,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-561","chapter","type-chapter","status-publish","hentry"],"part":376,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/561","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":10,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/561\/revisions"}],"predecessor-version":[{"id":3613,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/561\/revisions\/3613"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/parts\/376"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapters\/561\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=561"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/chapter-type?post=561"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=561"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=561"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}