{"id":345,"date":"2021-06-04T00:05:47","date_gmt":"2021-06-04T00:05:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/divide-monomials\/"},"modified":"2022-01-04T00:58:02","modified_gmt":"2022-01-04T00:58:02","slug":"divide-monomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/divide-monomials\/","title":{"raw":"8.5: Dividing Polynomials by a Monomial","rendered":"8.5: Dividing Polynomials by a Monomial"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide monomials<\/li>\r\n \t<li>Divide polynomials by monomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Quotient<\/strong>: the result of dividing<\/li>\r\n \t<li><strong>Quotient Property of Exponents<\/strong>:\u00a0to divide two terms with the same base, subtract the exponents and keep the common base<\/li>\r\n \t<li><strong>Dividend<\/strong>: the expression being divided<\/li>\r\n \t<li><strong>Divisor<\/strong>: the expression dividing into the dividend<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Dividing by a Monomial<\/h2>\r\nIn a previous chapter we learned about the properties of exponents. In particular, we learned that to divide two terms with the same base, we subtract the exponents and keep the common base:\r\n<p style=\"text-align: center;\">[latex]\\frac{x^m}{x^n}=x^{m-n}[\/latex]<\/p>\r\nWe will now use this <em><strong>quotient property of exponents<\/strong><\/em> to divide two monomials.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the quotient:\r\n<ol>\r\n \t<li>[latex]x^7\\div x^4[\/latex]<\/li>\r\n \t<li>[latex]\\frac{y^5}{y^4}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{n^8}{n^8}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{z^2}{z^5}[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]x^7\\div x^4=x^{7-4}=x^3[\/latex]\u00a0 \u00a0Keep the common base, [latex]x[\/latex], and subtract the exponents.<\/li>\r\n \t<li>[latex]\\frac{y^5}{y^4}=y^{5-4}=y^1=y[\/latex]\u00a0 \u00a0Keep the common base, [latex]y[\/latex], and subtract the exponents.<\/li>\r\n \t<li>[latex]\\frac{n^8}{n^8}=n^{8-8}=n^0=1[\/latex]\u00a0 \u00a0Remember that [latex]x^0=1[\/latex] for all [latex]x\\ne0[\/latex]<\/li>\r\n \t<li>[latex]\\frac{z^2}{z^5}=z^{2-5}=z^{-3}=\\frac{1}{z^3}[\/latex]\u00a0 Remember that a negative exponent on the numerator becomes a positive exponent on the denominator: [latex]x^{-n}=\\frac{1}{n}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the quotient:\r\n<ol>\r\n \t<li>[latex]x^6\\div x^2[\/latex]<\/li>\r\n \t<li>[latex]\\frac{y^8}{y^5}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{n^3}{n^3}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{z^4}{z^7}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm066\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm066\"]\r\n<ol>\r\n \t<li>[latex]x^6\\div x^2=x^4[\/latex]<\/li>\r\n \t<li>[latex]\\frac{y^8}{y^5}=y^3[\/latex]<\/li>\r\n \t<li>[latex]\\frac{n^3}{n^3}=1[\/latex]<\/li>\r\n \t<li>[latex]\\frac{z^4}{z^7}=\\frac{1}{z^3}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTechnically, any divisor cannot equal zero, as division by zero is undefined. Also, if we get an answer of [latex]x^0[\/latex], the answer will always be [latex]1[\/latex], provided that [latex]x\\ne0[\/latex].\u00a0 [latex]0^0[\/latex] is undefined.\u00a0 Notice that dividing two monomials does not always result in a monomial.\u00a0 For example,\u00a0[latex]\\frac{z^4}{z^7}=\\frac{1}{z^3}[\/latex] does not result in a monomial. Remember that monomials cannot have negative exponents; the exponent must be a whole number.\r\n\r\nWhen there are coefficients attached to the variables, we divide the coefficients and divide the variables.\r\n<div class=\"textbox exercises\">\r\n<h3>\u00a0EXAMPLE<\/h3>\r\nFind the quotient: [latex]56{x}^{5}\\div 7{x}^{2}[\/latex]\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168466176948\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\"><\/td>\r\n<td style=\"height: 15px;\">[latex]56{x}^{5}\\div 7{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.2px;\">\r\n<td style=\"height: 15.2px;\">Rewrite as a fraction.<\/td>\r\n<td style=\"height: 15.2px;\">[latex]{\\dfrac{56{x}^{5}}{7{x}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 74px;\">\r\n<td style=\"height: 74px;\">Use fraction multiplication to separate the number\r\n\r\npart from the variable part.<\/td>\r\n<td style=\"height: 74px;\">[latex]{\\dfrac{56}{7}}\\cdot {\\dfrac{{x}^{5}}{{x}^{2}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Use the Quotient Property: keep the base, subtract the exponents<\/td>\r\n<td style=\"height: 15px;\">[latex]8{x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Answer<\/h4>\r\n[latex]56{x}^{5}\\div 7{x}^{2}=8{x}^{3}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n1. Find the quotient: [latex]63{x}^{8}\\div 9{x}^{4}[\/latex]\r\n\r\n[reveal-answer q=\"980543\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"980543\"]\r\n\r\n[latex]7{x}^{4}[\/latex]<sup>\u00a0<\/sup>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n2. Find the quotient: [latex]96{y}^{11}\\div 6{y}^{8}[\/latex]\r\n\r\n[reveal-answer q=\"245633\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"245633\"]\r\n\r\n[latex]16{y}^{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]186447[\/ohm_question]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Dividing a Polynomial by a Monomial<\/h2>\r\nThe distributive property states that we can distribute a factor that is being multiplied by a sum or difference: [latex]a(b+c)=ab+ac[\/latex].\u00a0 If the term being multiplied is a fraction, [latex]\\frac{1}{a}[\/latex], the distributive property tells us:\r\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}&amp;\\;\\;\\;\\; \\\\ \\frac{1}{a}(b+c)&amp;=\\frac{1}{a}\\cdot b+\\frac{1}{a}\\cdot c\\end{aligned}\\end{equation}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">But, [latex]\\frac{1}{a}(b+c)=\\frac{b+c}{a}[\/latex] and [latex]\\frac{1}{a}\\cdot b+\\frac{1}{a}\\cdot c=\\frac{b}{a}+\\frac{c}{a}[\/latex]\u00a0by multiplication of fractions.<\/p>\r\n<p style=\"text-align: center;\">So,\u00a0[latex]\\frac{b+c}{a}=\\frac{b}{a}+\\frac{c}{a}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">In other words, we can distribute a <strong><em>divisor<\/em><\/strong> that is being divided into a sum or difference.<\/p>\r\nIn this arithmetic example, we can add all the terms in the numerator, then divide by [latex]2[\/latex].\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 we can\u00a0first divide each term by [latex]2[\/latex], 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 the same result. The second way is helpful when we can't combine like terms in the numerator, as in a polynomial divided by a monomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]\r\n<h4>Solution<\/h4>\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<\/div>\r\nThe distributive property can be extended to any number of terms, so the next example applies the same ideas to divide a trinomial by a monomial. We can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. Remember that simplifying with exponents includes rewriting negative exponents in the numerator as positive exponents in the denominator. Also remember to pay attention to the signs of the terms.\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<h4>Solution<\/h4>\r\nDivide 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 [latex]y[\/latex] in the numerator, so division is only applied to the numbers [latex]18[\/latex] and [latex]-6[\/latex]. Also, 27 doesn't divide exactly 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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7873[\/ohm_question]\r\n\r\n<\/div>\r\nNo matter the number of terms in the polynomial, we can use the distributive property to divide by a monomial.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDivide [latex]6x^6-3x^4+9x^2-7[\/latex] by [latex]-3x^3[\/latex]\r\n<h4>Solution<\/h4>\r\nWrite as division:\r\n\r\n[latex]\\frac{6x^6-3x^4+9x^2-7}{-3x^3}[\/latex]\r\n\r\nDistribute the monomial to each term in the polynomial:\r\n\r\n[latex]\\frac{6x^6}{-3x^3}-\\frac{3x^4}{-3x^3}+\\frac{9x^2}{-3x^3}-\\frac{7}{-3x^3}[\/latex]\r\n\r\nSimplify:\r\n\r\n[latex]-2x^3+x-3x^{-1}+\\frac{7}{3x^3}[\/latex]\r\n\r\nWrite the negative exponent as a positive exponent on the denominator:\r\n\r\n[latex]-2x^3+x-\\frac{3}{x}+\\frac{7}{3x^3}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{6x^6-3x^4+9x^2-7}{-3x^3}=-2x^3+x-\\frac{3}{x}+\\frac{7}{3x^3}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDivide:\u00a0 [latex]\\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}[\/latex]\r\n<h4>Solution<\/h4>\r\n[latex]\\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}[\/latex]\r\n\r\nDistribute [latex]-12x^2[\/latex] to each term of the polynomial:\r\n\r\n[latex]\\frac{24x^8}{-12x^2}+\\frac{36x^7}{-12x^2}-\\frac{-12x^4}{-12x^2}-\\frac{-60x^3}{-12x^2}+\\frac{6x}{-12x^2}[\/latex]\r\n\r\nSimplify:\r\n\r\n[latex]-2x^6-3x^5+x^2+5x-\\frac{1}{2}x^{-1}[\/latex]\r\n\r\nWrite the negative exponent as a positive exponent on the denominator:\r\n\r\n[latex]-2x^6-3x^5+x^2+5x-\\frac{1}{2x}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}=-2x^6-3x^5+x^2+5x-\\frac{1}{2x}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n1. Divide:\u00a0\u00a0[latex]\\frac{36x^6-16x^4-24x^3+6x}{-4x^2}[\/latex]\r\n\r\n[reveal-answer q=\"hjm584\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm584\"][latex]-9x^4+4x^2+6x-\\frac{3}{2x}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n2.\u00a0\u00a0Divide:\u00a0\u00a0[latex]\\frac{42x^7-14x^5+21x^4-35x^3+6x}{7x^3}[\/latex]\r\n\r\n[reveal-answer q=\"hjm920\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm920\"][latex]6x^4-2x^2+3x-5+\\frac{6}{7x^2}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n3.\u00a0\u00a0Divide:\u00a0\u00a0[latex]\\frac{-42x^9-24x^7+9x^4-36x^3-x}{-6x^4}[\/latex]\r\n\r\n[reveal-answer q=\"hjm617\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm617\"][latex]7x^5+4x^3-\\frac{3}{2}+\\frac{6}{x}+\\frac{1}{6x^3}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide monomials<\/li>\n<li>Divide polynomials by monomials<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Quotient<\/strong>: the result of dividing<\/li>\n<li><strong>Quotient Property of Exponents<\/strong>:\u00a0to divide two terms with the same base, subtract the exponents and keep the common base<\/li>\n<li><strong>Dividend<\/strong>: the expression being divided<\/li>\n<li><strong>Divisor<\/strong>: the expression dividing into the dividend<\/li>\n<\/ul>\n<\/div>\n<h2>Dividing by a Monomial<\/h2>\n<p>In a previous chapter we learned about the properties of exponents. In particular, we learned that to divide two terms with the same base, we subtract the exponents and keep the common base:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{x^m}{x^n}=x^{m-n}[\/latex]<\/p>\n<p>We will now use this <em><strong>quotient property of exponents<\/strong><\/em> to divide two monomials.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the quotient:<\/p>\n<ol>\n<li>[latex]x^7\\div x^4[\/latex]<\/li>\n<li>[latex]\\frac{y^5}{y^4}[\/latex]<\/li>\n<li>[latex]\\frac{n^8}{n^8}[\/latex]<\/li>\n<li>[latex]\\frac{z^2}{z^5}[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]x^7\\div x^4=x^{7-4}=x^3[\/latex]\u00a0 \u00a0Keep the common base, [latex]x[\/latex], and subtract the exponents.<\/li>\n<li>[latex]\\frac{y^5}{y^4}=y^{5-4}=y^1=y[\/latex]\u00a0 \u00a0Keep the common base, [latex]y[\/latex], and subtract the exponents.<\/li>\n<li>[latex]\\frac{n^8}{n^8}=n^{8-8}=n^0=1[\/latex]\u00a0 \u00a0Remember that [latex]x^0=1[\/latex] for all [latex]x\\ne0[\/latex]<\/li>\n<li>[latex]\\frac{z^2}{z^5}=z^{2-5}=z^{-3}=\\frac{1}{z^3}[\/latex]\u00a0 Remember that a negative exponent on the numerator becomes a positive exponent on the denominator: [latex]x^{-n}=\\frac{1}{n}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the quotient:<\/p>\n<ol>\n<li>[latex]x^6\\div x^2[\/latex]<\/li>\n<li>[latex]\\frac{y^8}{y^5}[\/latex]<\/li>\n<li>[latex]\\frac{n^3}{n^3}[\/latex]<\/li>\n<li>[latex]\\frac{z^4}{z^7}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm066\">Show Answer<\/span><\/p>\n<div id=\"qhjm066\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]x^6\\div x^2=x^4[\/latex]<\/li>\n<li>[latex]\\frac{y^8}{y^5}=y^3[\/latex]<\/li>\n<li>[latex]\\frac{n^3}{n^3}=1[\/latex]<\/li>\n<li>[latex]\\frac{z^4}{z^7}=\\frac{1}{z^3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>Technically, any divisor cannot equal zero, as division by zero is undefined. Also, if we get an answer of [latex]x^0[\/latex], the answer will always be [latex]1[\/latex], provided that [latex]x\\ne0[\/latex].\u00a0 [latex]0^0[\/latex] is undefined.\u00a0 Notice that dividing two monomials does not always result in a monomial.\u00a0 For example,\u00a0[latex]\\frac{z^4}{z^7}=\\frac{1}{z^3}[\/latex] does not result in a monomial. Remember that monomials cannot have negative exponents; the exponent must be a whole number.<\/p>\n<p>When there are coefficients attached to the variables, we divide the coefficients and divide the variables.<\/p>\n<div class=\"textbox exercises\">\n<h3>\u00a0EXAMPLE<\/h3>\n<p>Find the quotient: [latex]56{x}^{5}\\div 7{x}^{2}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168466176948\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\"><\/td>\n<td style=\"height: 15px;\">[latex]56{x}^{5}\\div 7{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.2px;\">\n<td style=\"height: 15.2px;\">Rewrite as a fraction.<\/td>\n<td style=\"height: 15.2px;\">[latex]{\\dfrac{56{x}^{5}}{7{x}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 74px;\">\n<td style=\"height: 74px;\">Use fraction multiplication to separate the number<\/p>\n<p>part from the variable part.<\/td>\n<td style=\"height: 74px;\">[latex]{\\dfrac{56}{7}}\\cdot {\\dfrac{{x}^{5}}{{x}^{2}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Use the Quotient Property: keep the base, subtract the exponents<\/td>\n<td style=\"height: 15px;\">[latex]8{x}^{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Answer<\/h4>\n<p>[latex]56{x}^{5}\\div 7{x}^{2}=8{x}^{3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p>1. Find the quotient: [latex]63{x}^{8}\\div 9{x}^{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q980543\">Show Solution<\/span><\/p>\n<div id=\"q980543\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]7{x}^{4}[\/latex]<sup>\u00a0<\/sup><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>2. Find the quotient: [latex]96{y}^{11}\\div 6{y}^{8}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q245633\">Show Solution<\/span><\/p>\n<div id=\"q245633\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]16{y}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm186447\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=186447&theme=oea&iframe_resize_id=ohm186447&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\n<h2>Dividing a Polynomial by a Monomial<\/h2>\n<p>The distributive property states that we can distribute a factor that is being multiplied by a sum or difference: [latex]a(b+c)=ab+ac[\/latex].\u00a0 If the term being multiplied is a fraction, [latex]\\frac{1}{a}[\/latex], the distributive property tells us:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{equation}\\begin{aligned}&\\;\\;\\;\\; \\\\ \\frac{1}{a}(b+c)&=\\frac{1}{a}\\cdot b+\\frac{1}{a}\\cdot c\\end{aligned}\\end{equation}[\/latex]<\/p>\n<p style=\"text-align: center;\">But, [latex]\\frac{1}{a}(b+c)=\\frac{b+c}{a}[\/latex] and [latex]\\frac{1}{a}\\cdot b+\\frac{1}{a}\\cdot c=\\frac{b}{a}+\\frac{c}{a}[\/latex]\u00a0by multiplication of fractions.<\/p>\n<p style=\"text-align: center;\">So,\u00a0[latex]\\frac{b+c}{a}=\\frac{b}{a}+\\frac{c}{a}[\/latex]<\/p>\n<p style=\"text-align: left;\">In other words, we can distribute a <strong><em>divisor<\/em><\/strong> that is being divided into a sum or difference.<\/p>\n<p>In this arithmetic example, we can add all the terms in the numerator, then divide by [latex]2[\/latex].<\/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 we can\u00a0first divide each term by [latex]2[\/latex], 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 the same result. The second way is helpful when we can&#8217;t combine like terms in the numerator, as in a polynomial divided by a monomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]<\/p>\n<h4>Solution<\/h4>\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<p>The distributive property can be extended to any number of terms, so the next example applies the same ideas to divide a trinomial by a monomial. We can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. Remember that simplifying with exponents includes rewriting negative exponents in the numerator as positive exponents in the denominator. Also remember to pay attention to the signs of the terms.<\/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<h4>Solution<\/h4>\n<p>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 [latex]y[\/latex] in the numerator, so division is only applied to the numbers [latex]18[\/latex] and [latex]-6[\/latex]. Also, 27 doesn&#8217;t divide exactly 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 class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7873\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7873&theme=oea&iframe_resize_id=ohm7873&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>No matter the number of terms in the polynomial, we can use the distributive property to divide by a monomial.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Divide [latex]6x^6-3x^4+9x^2-7[\/latex] by [latex]-3x^3[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Write as division:<\/p>\n<p>[latex]\\frac{6x^6-3x^4+9x^2-7}{-3x^3}[\/latex]<\/p>\n<p>Distribute the monomial to each term in the polynomial:<\/p>\n<p>[latex]\\frac{6x^6}{-3x^3}-\\frac{3x^4}{-3x^3}+\\frac{9x^2}{-3x^3}-\\frac{7}{-3x^3}[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p>[latex]-2x^3+x-3x^{-1}+\\frac{7}{3x^3}[\/latex]<\/p>\n<p>Write the negative exponent as a positive exponent on the denominator:<\/p>\n<p>[latex]-2x^3+x-\\frac{3}{x}+\\frac{7}{3x^3}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{6x^6-3x^4+9x^2-7}{-3x^3}=-2x^3+x-\\frac{3}{x}+\\frac{7}{3x^3}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Divide:\u00a0 [latex]\\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>[latex]\\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}[\/latex]<\/p>\n<p>Distribute [latex]-12x^2[\/latex] to each term of the polynomial:<\/p>\n<p>[latex]\\frac{24x^8}{-12x^2}+\\frac{36x^7}{-12x^2}-\\frac{-12x^4}{-12x^2}-\\frac{-60x^3}{-12x^2}+\\frac{6x}{-12x^2}[\/latex]<\/p>\n<p>Simplify:<\/p>\n<p>[latex]-2x^6-3x^5+x^2+5x-\\frac{1}{2}x^{-1}[\/latex]<\/p>\n<p>Write the negative exponent as a positive exponent on the denominator:<\/p>\n<p>[latex]-2x^6-3x^5+x^2+5x-\\frac{1}{2x}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{24x^8+36x^7-12x^4-60x^3+6x}{-12x^2}=-2x^6-3x^5+x^2+5x-\\frac{1}{2x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>1. Divide:\u00a0\u00a0[latex]\\frac{36x^6-16x^4-24x^3+6x}{-4x^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm584\">Show Answer<\/span><\/p>\n<div id=\"qhjm584\" class=\"hidden-answer\" style=\"display: none\">[latex]-9x^4+4x^2+6x-\\frac{3}{2x}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>2.\u00a0\u00a0Divide:\u00a0\u00a0[latex]\\frac{42x^7-14x^5+21x^4-35x^3+6x}{7x^3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm920\">Show Answer<\/span><\/p>\n<div id=\"qhjm920\" class=\"hidden-answer\" style=\"display: none\">[latex]6x^4-2x^2+3x-5+\\frac{6}{7x^2}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>3.\u00a0\u00a0Divide:\u00a0\u00a0[latex]\\frac{-42x^9-24x^7+9x^4-36x^3-x}{-6x^4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm617\">Show Answer<\/span><\/p>\n<div id=\"qhjm617\" class=\"hidden-answer\" style=\"display: none\">[latex]7x^5+4x^3-\\frac{3}{2}+\\frac{6}{x}+\\frac{1}{6x^3}[\/latex]<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/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-345\">\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>Try it: hjm584; hjm920; hjm617; hjm066. Examples 1; 5; 6. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Question ID: 146014, 146148. <strong>Authored 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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Revised and adapted: Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":23485,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Revised and adapted: Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Question ID: 146014, 146148\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Try it: hjm584; hjm920; hjm617; hjm066. Examples 1; 5; 6\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"85a435f83bbe4b7c84f32b954269e6c1, 5e75bf6ceab34a3184ed6470387648e4","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-345","chapter","type-chapter","status-publish","hentry"],"part":663,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/345","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/345\/revisions"}],"predecessor-version":[{"id":2045,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/345\/revisions\/2045"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/663"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/345\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=345"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=345"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=345"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}