{"id":4745,"date":"2017-06-07T18:56:22","date_gmt":"2017-06-07T18:56:22","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-divide-polynomials\/"},"modified":"2017-08-15T20:40:48","modified_gmt":"2017-08-15T20:40:48","slug":"read-divide-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-divide-polynomials\/","title":{"raw":"Divide Polynomials","rendered":"Divide Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Divide a binomial by a monomial<\/li>\r\n \t<li>Divide a trinomial by a monomial<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe fourth arithmetic operation is division, the inverse of multiplication. Division of polynomials isn\u2019t much different from division of numbers. In the exponential section,\u00a0you were asked to simplify expressions such as:\u00a0[latex]\\displaystyle\\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]. This expression is the division of two monomials.\u00a0To simplify it, we\u00a0divided\u00a0the coefficients and then divided the variables. In this section we will add another layer to this idea by dividing polynomials by monomials, and by binomials.\r\n<h2>Divide a polynomial by a monomial<\/h2>\r\nThe distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a <em>divisor<\/em> that is being divided into a sum or difference. In this example, you can add all the terms in the numerator, then divide by 2.\r\n<p style=\"text-align: center\">[latex]\\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\r\nOr you can\u00a0first divide each term by 2, then simplify the result.\r\n<p style=\"text-align: center\">[latex] \\frac{8}{2}+\\frac{4}{2}+\\frac{10}{2}=4+2+5=11[\/latex]<\/p>\r\nEither way gives you the same result. The second way is helpful when you can't combine like terms in the numerator. \u00a0Let\u2019s try something similar with a binomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]\r\n[reveal-answer q=\"641821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641821\"]\r\n\r\nDistribute [latex]3a^2[\/latex]<i>\u00a0<\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]\r\n\r\n[latex]\\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]\r\n\r\nDivide each term, a monomial divided by another monomial.\r\n\r\n[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]\r\n\r\nRewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.\r\n\r\n[latex]3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide. [latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]\r\n[reveal-answer q=\"324719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324719\"]Divide each term in the polynomial by the monomial.\r\n<p style=\"text-align: center\">[latex] \\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\r\nNote how the term[latex]-\\frac{18}{-6y}[\/latex] does not have a <em>y<\/em> in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, 27 doesn't divide nicely by [latex]-6[\/latex], so we are left with a fraction as the coefficient on the [latex]y^3[\/latex] term.\r\n\r\nSimplify.\r\n<p style=\"text-align: center\">[latex] -\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Summary<\/h2>\r\nTo divide a monomial by a monomial, divide the coefficients (or simplify them as you would a fraction) and divide the variables with like bases by subtracting their exponents. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Be sure to watch the signs! Final answers should be written without any negative exponents.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Divide a binomial by a monomial<\/li>\n<li>Divide a trinomial by a monomial<\/li>\n<\/ul>\n<\/div>\n<p>The fourth arithmetic operation is division, the inverse of multiplication. Division of polynomials isn\u2019t much different from division of numbers. In the exponential section,\u00a0you were asked to simplify expressions such as:\u00a0[latex]\\displaystyle\\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]. This expression is the division of two monomials.\u00a0To simplify it, we\u00a0divided\u00a0the coefficients and then divided the variables. In this section we will add another layer to this idea by dividing polynomials by monomials, and by binomials.<\/p>\n<h2>Divide a polynomial by a monomial<\/h2>\n<p>The distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a <em>divisor<\/em> that is being divided into a sum or difference. In this example, you can add all the terms in the numerator, then divide by 2.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\n<p>Or you can\u00a0first divide each term by 2, then simplify the result.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{8}{2}+\\frac{4}{2}+\\frac{10}{2}=4+2+5=11[\/latex]<\/p>\n<p>Either way gives you the same result. The second way is helpful when you can&#8217;t combine like terms in the numerator. \u00a0Let\u2019s try something similar with a binomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q641821\">Show Solution<\/span><\/p>\n<div id=\"q641821\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute [latex]3a^2[\/latex]<i>\u00a0<\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]<\/p>\n<p>[latex]\\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]<\/p>\n<p>Divide each term, a monomial divided by another monomial.<\/p>\n<p>[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.<\/p>\n<p>[latex]3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324719\">Show Solution<\/span><\/p>\n<div id=\"q324719\" class=\"hidden-answer\" style=\"display: none\">Divide each term in the polynomial by the monomial.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\n<p>Note how the term[latex]-\\frac{18}{-6y}[\/latex] does not have a <em>y<\/em> in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, 27 doesn&#8217;t divide nicely by [latex]-6[\/latex], so we are left with a fraction as the coefficient on the [latex]y^3[\/latex] term.<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>Summary<\/h2>\n<p>To divide a monomial by a monomial, divide the coefficients (or simplify them as you would a fraction) and divide the variables with like bases by subtracting their exponents. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Be sure to watch the signs! Final answers should be written without any negative exponents.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4745\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Rxds7Q_UTeo\">https:\/\/youtu.be\/Rxds7Q_UTeo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Divide a Trinomial by a Binomial Using Long Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KUPFg__Djzw\">https:\/\/youtu.be\/KUPFg__Djzw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 6: Divide a Polynomial by a Degree Two Binomial Using Long Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P6OTbUf8f60\">https:\/\/youtu.be\/P6OTbUf8f60<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">Lumen Learning authored content<\/div><ul class=\"citation-list\"><li>Screenshot I&#039;m Not Allergic to Long Division. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"lumen\",\"description\":\"Screenshot I\\'m Not Allergic to Long 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