{"id":16260,"date":"2019-10-02T20:23:07","date_gmt":"2019-10-02T20:23:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/4-2-5-dividing-by-a-monomial\/"},"modified":"2020-10-22T09:29:41","modified_gmt":"2020-10-22T09:29:41","slug":"4-2-5-dividing-by-a-monomial","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/4-2-5-dividing-by-a-monomial\/","title":{"raw":"12.3.b - Dividing Polynomials","rendered":"12.3.b &#8211; Dividing Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide polynomials by monomials<\/li>\r\n \t<li>Divide polynomials by binomials<\/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.\u00a0 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 [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 you 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 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[\/latex] and [latex]-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<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7873[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow, we ask you to think about what would happen if you were given a quotient like this to simplify:\u00a0[latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y+3}[\/latex]. \u00a0You may be tempted to divide each term of\u00a0[latex] {27{{y}^{4}}+6{{y}^{2}}-18}[\/latex] individually by [latex]-6y[\/latex], then [latex]3[\/latex]. \u00a0This would go against the order of operations because the division sign is a grouping symbol, and the addition in the denominator cannot be simplified anymore. The result is that\u00a0no further operations can be performed with the tools we know. \u00a0We can, however, call into use a tool that you may have learned in gradeschool: long division.\r\n<h2 id=\"title1\">Polynomial Long Division<\/h2>\r\nRecall how you can use long division to divide two whole numbers, say [latex]900[\/latex] divided by [latex]37[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064125\/image007.jpg\" alt=\"The dividend in 900 and the divisor is 37.\" width=\"71\" height=\"25\" \/>\r\nFirst, you would think about how many [latex]37[\/latex]s are in [latex]90[\/latex], as [latex]9[\/latex] is too small. (<i>Note: <\/i>you could also think, how many [latex]40[\/latex]s are there in [latex]90[\/latex].)\r\n\r\n<img class=\"aligncenter wp-image-2251 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223536\/Screen-Shot-2016-03-28-at-3.35.17-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.35.17 PM\" width=\"97\" height=\"77\" \/>\r\nThere are two [latex]37[\/latex]s in [latex]90[\/latex], so write [latex]2[\/latex] above the last digit of \u00a0[latex]90[\/latex]. Two [latex]37[\/latex]s is [latex]74[\/latex]; write that product below the [latex]90[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image008.jpg\" alt=\"\" width=\"68\" height=\"76\" \/>\r\nSubtract: [latex]90\u201374[\/latex] is [latex]16[\/latex]. (If the result is larger than the divisor, [latex]37[\/latex], then you need to use a larger number for the quotient.)\r\n\r\n<img class=\"aligncenter wp-image-2252\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223630\/Screen-Shot-2016-03-28-at-3.36.06-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.36.06 PM\" width=\"84\" height=\"86\" \/>\r\n\r\nBring down the next digit [latex](0)[\/latex] and consider how many [latex]37[\/latex]s are in [latex]160[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image009.jpg\" alt=\"\" width=\"64\" height=\"93\" \/>\r\nThere are four [latex]37[\/latex]s in [latex]160[\/latex], so write the [latex]4[\/latex] next to the two in the quotient. Four [latex]37[\/latex]s is [latex]148[\/latex]; write that product below the [latex]160[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image010.jpg\" alt=\"\" width=\"62\" height=\"104\" \/>\r\nSubtract: [latex]160\u2013148[\/latex] is [latex]12[\/latex]. This is less than [latex]37[\/latex] so the [latex]4[\/latex] is correct. Since there are no more digits in the dividend to bring down, you\u2019re done.\r\n\r\nThe final answer is [latex]24[\/latex] R[latex]12[\/latex], or [latex]24\\frac{12}{37}[\/latex]. You can check this by multiplying the quotient (without the remainder) by the divisor, and then adding in the remainder. The result should be the dividend:\r\n<p style=\"text-align: center\">[latex]24\\cdot37+12=888+12=900[\/latex]<\/p>\r\nTo divide polynomials, use the same process. This example shows how to do this when dividing by a <strong>binomial<\/strong>.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide:\u00a0[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}[\/latex]\r\n[reveal-answer q=\"455187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455187\"]How many <i>x<\/i>\u2019s are there in [latex]x^{2}[\/latex]? That is, what is [latex] \\frac{{{x}^{2}}}{x}[\/latex]?\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image011.jpg\" alt=\"\" width=\"138\" height=\"21\" \/>\r\n\r\n[latex] \\frac{{{x}^{2}}}{x}=x[\/latex]<i>. <\/i>Put <i>x<\/i> in the quotient above the [latex]-4x[\/latex]<i>\u00a0<\/i>term. (These are like terms, which helps to organize the problem.)\r\n\r\nWrite the product of the divisor and the part of the quotient you just found under the dividend. Since [latex]x\\left(x+2\\right)=x^{2}+2x[\/latex],\u00a0write this underneath, and get ready to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image012.jpg\" alt=\"\" width=\"134\" height=\"58\" \/>\r\n\r\nRewrite [latex]\u2013\\left(x^{2} + 2x\\right)[\/latex]\u00a0as its opposite [latex]\u2013x^{2}\u20132x[\/latex]\u00a0so that you can add the opposite. (Adding the opposite is the same as subtracting, and it is easier to do.)\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image013.jpg\" alt=\"\" width=\"135\" height=\"55\" \/>\r\n\r\nAdd\u00a0[latex]-x^{2}[\/latex] to [latex]x^{2}[\/latex], and [latex]-2x[\/latex] to [latex]-4x[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064128\/image014.jpg\" alt=\"\" width=\"136\" height=\"69\" \/>\r\n\r\nBring down [latex]-12[\/latex].\r\n\r\n<img class=\"aligncenter wp-image-2253\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232007\/Screen-Shot-2016-03-28-at-4.19.50-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.19.50 PM\" width=\"152\" height=\"83\" \/>\r\n\r\nRepeat the process. How many times does <i>x<\/i> go into [latex]-6x[\/latex]? In other words, what is [latex] \\frac{-6x}{x}[\/latex]?\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image015.jpg\" alt=\"\" width=\"126\" height=\"64\" \/>\r\n\r\nSince [latex] \\frac{-6x}{x}=-6[\/latex], write [latex]-6[\/latex] in the quotient. Multiply [latex]-6[\/latex] and [latex]x+2[\/latex]\u00a0and prepare to subtract the product.\r\n\r\n<img class=\"aligncenter size-full wp-image-2254\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232512\/Screen-Shot-2016-03-28-at-4.24.52-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.24.52 PM\" width=\"165\" height=\"113\" \/>\r\n\r\nRewrite [latex]\u2013\\left(-6x\u201312\\right)[\/latex] as [latex]6x+12[\/latex], so that you can add the opposite.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image016.jpg\" alt=\"\" width=\"127\" height=\"86\" \/>\r\n\r\nAdd. In this case, there is no remainder, so you\u2019re done.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image017.jpg\" alt=\"\" width=\"134\" height=\"102\" \/>\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}=x\u20136[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nCheck this by multiplying:\r\n<p style=\"text-align: center\">[latex]\\left(x-6\\right)\\left(x+2\\right)=x^{2}+2x-6x-12=x^{2}-4x-12[\/latex]<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRy It<\/h3>\r\n[ohm_question]3855[\/ohm_question]\r\n\r\n<\/div>\r\nPolynomial long division involves many steps. \u00a0Hopefully this video will help you determine what step to do next when you are using polynomial long division.\r\n\r\nhttps:\/\/youtu.be\/KUPFg__Djzw\r\n\r\nLet\u2019s try another example. In this example, a term is \u201cmissing\u201d from the dividend.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide: [latex]\\frac{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}[\/latex]\r\n[reveal-answer q=\"523374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"523374\"]In setting up this problem, notice that there is an [latex]x^{3}[\/latex]\u00a0term but no [latex]x^{2}[\/latex]\u00a0term. Add [latex]0x^{2}[\/latex]\u00a0as a \u201cplace holder\u201d for this term. (Since 0 times anything is 0, you\u2019re not changing the value of the dividend.)\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image018.jpg\" alt=\"\" width=\"153\" height=\"18\" \/>\r\n\r\nFocus on the first terms again: how many <i>x<\/i>\u2019s are there in [latex]x^{3}[\/latex]? Since [latex] \\frac{{{x}^{3}}}{x}=x^{2}[\/latex], put [latex]x^{2}[\/latex]\u00a0in the quotient.\r\n\r\nMultiply [latex]x^{2}\\left(x\u20133\\right)=x^{3}\u20133x^{2}[\/latex], write this underneath the dividend, and prepare to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064131\/image019.jpg\" alt=\"\" width=\"153\" height=\"49\" \/>\r\n\r\nRewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[\/latex]. Then add.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064132\/image020.jpg\" alt=\"\" width=\"153\" height=\"59\" \/>\r\n\r\nBring down the rest of the expression in the dividend. It\u2019s helpful to bring down <i>all<\/i> of the remaining terms.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064133\/image021.jpg\" alt=\"\" width=\"153\" height=\"59\" \/>\r\n\r\nNow, repeat the process with the remaining expression, [latex]3x^{2}-6x\u201310[\/latex], as the dividend.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image022.jpg\" alt=\"\" width=\"153\" height=\"79\" \/>\r\n\r\nRemember to watch the signs!\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image023.jpg\" alt=\"\" width=\"153\" height=\"89\" \/>\r\n\r\nHow many [latex]x[\/latex]\u2019s are there in [latex]3[\/latex]x? Since there are [latex]3[\/latex], multiply [latex]3\\left(x\u20133\\right)=3x\u20139[\/latex], write this underneath the dividend, and prepare to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064135\/image024.jpg\" alt=\"\" width=\"157\" height=\"118\" \/>\r\n\r\nContinue until the <strong>degree<\/strong> of the remainder is <i>less <\/i>than the degree of the divisor. In this case the degree of the remainder, [latex]-1[\/latex], is [latex]0[\/latex], which is less than the degree of [latex]x-3[\/latex], which is [latex]1[\/latex].\r\n\r\nAlso notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064221\/image025.jpg\" alt=\"\" width=\"153\" height=\"118\" \/>\r\nYou can write the remainder using the symbol R, or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.\r\n<h4>Answer<\/h4>\r\n[latex]\\begin{array}{r}{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}=x^{2}+3x+3+R-1,\\\\x^{2}+3x+3+\\frac{-1}{x-3}, \\text{ or }\\\\x^{2}+3x+3-\\frac{1}{x-3}\\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nCheck the result:\r\n<p style=\"text-align: center\">[latex]\\left(x\u20133\\right)\\left(x^{2}+3x+3\\right)\\,\\,\\,=\\,\\,\\,x\\left(x^{2}+3x+3\\right)\u20133\\left(x^{2}+3x+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}+3x^{2}+3x\u20133x^{2}\u20139x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,x^{3}\u20136x\u20139+\\left(-1\\right)\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u201310[\/latex]<\/p>\r\nThe video that follows shows another example of dividing a third degree trinomial by a first degree binomial.\u00a0 \u00a0Note the \"missing term\" and pay close attention to how we work with it.\r\n\r\nhttps:\/\/youtu.be\/Rxds7Q_UTeo\r\n\r\nThe process above works for dividing any polynomials no matter how many terms are in the divisor or the dividend and no matter what the degree of the polynomials are <em>as long as the degree of the dividend is greater than or equal to the degree of the divisor<\/em>. The main things to remember are:\r\n<ul>\r\n \t<li>When subtracting, be sure to subtract the whole expression, not just the first term. <i>This is very easy to forget, so be careful!<\/i><\/li>\r\n \t<li>Stop when the degree of the remainder is less than the degree of the dividend or when you have brought down all the terms in the dividend and your quotient extends to the right edge of the dividend.<\/li>\r\n<\/ul>\r\nThe last video example shows how to divide a third degree trinomial by a second degree binomial.\r\n\r\nhttps:\/\/youtu.be\/P6OTbUf8f60\r\n<h2>Summary<\/h2>\r\nTo 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. \u00a0Dividing polynomials by polynomials of more than one term can be done using a process very much like long division of whole numbers. You must be careful to subtract entire expressions, not just the first term. Stop when the degree of the remainder is less than the degree of the divisor. The remainder can be written using R notation, or as a fraction added to the quotient with the remainder in the numerator and the divisor in the denominator.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide polynomials by monomials<\/li>\n<li>Divide polynomials by binomials<\/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.\u00a0 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 [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 you 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 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[\/latex] and [latex]-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<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>&nbsp;<\/p>\n<p>Now, we ask you to think about what would happen if you were given a quotient like this to simplify:\u00a0[latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y+3}[\/latex]. \u00a0You may be tempted to divide each term of\u00a0[latex]{27{{y}^{4}}+6{{y}^{2}}-18}[\/latex] individually by [latex]-6y[\/latex], then [latex]3[\/latex]. \u00a0This would go against the order of operations because the division sign is a grouping symbol, and the addition in the denominator cannot be simplified anymore. The result is that\u00a0no further operations can be performed with the tools we know. \u00a0We can, however, call into use a tool that you may have learned in gradeschool: long division.<\/p>\n<h2 id=\"title1\">Polynomial Long Division<\/h2>\n<p>Recall how you can use long division to divide two whole numbers, say [latex]900[\/latex] divided by [latex]37[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064125\/image007.jpg\" alt=\"The dividend in 900 and the divisor is 37.\" width=\"71\" height=\"25\" \/><br \/>\nFirst, you would think about how many [latex]37[\/latex]s are in [latex]90[\/latex], as [latex]9[\/latex] is too small. (<i>Note: <\/i>you could also think, how many [latex]40[\/latex]s are there in [latex]90[\/latex].)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2251\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223536\/Screen-Shot-2016-03-28-at-3.35.17-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.35.17 PM\" width=\"97\" height=\"77\" \/><br \/>\nThere are two [latex]37[\/latex]s in [latex]90[\/latex], so write [latex]2[\/latex] above the last digit of \u00a0[latex]90[\/latex]. Two [latex]37[\/latex]s is [latex]74[\/latex]; write that product below the [latex]90[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image008.jpg\" alt=\"\" width=\"68\" height=\"76\" \/><br \/>\nSubtract: [latex]90\u201374[\/latex] is [latex]16[\/latex]. (If the result is larger than the divisor, [latex]37[\/latex], then you need to use a larger number for the quotient.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2252\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223630\/Screen-Shot-2016-03-28-at-3.36.06-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.36.06 PM\" width=\"84\" height=\"86\" \/><\/p>\n<p>Bring down the next digit [latex](0)[\/latex] and consider how many [latex]37[\/latex]s are in [latex]160[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image009.jpg\" alt=\"\" width=\"64\" height=\"93\" \/><br \/>\nThere are four [latex]37[\/latex]s in [latex]160[\/latex], so write the [latex]4[\/latex] next to the two in the quotient. Four [latex]37[\/latex]s is [latex]148[\/latex]; write that product below the [latex]160[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image010.jpg\" alt=\"\" width=\"62\" height=\"104\" \/><br \/>\nSubtract: [latex]160\u2013148[\/latex] is [latex]12[\/latex]. This is less than [latex]37[\/latex] so the [latex]4[\/latex] is correct. Since there are no more digits in the dividend to bring down, you\u2019re done.<\/p>\n<p>The final answer is [latex]24[\/latex] R[latex]12[\/latex], or [latex]24\\frac{12}{37}[\/latex]. You can check this by multiplying the quotient (without the remainder) by the divisor, and then adding in the remainder. The result should be the dividend:<\/p>\n<p style=\"text-align: center\">[latex]24\\cdot37+12=888+12=900[\/latex]<\/p>\n<p>To divide polynomials, use the same process. This example shows how to do this when dividing by a <strong>binomial<\/strong>.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide:\u00a0[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455187\">Show Solution<\/span><\/p>\n<div id=\"q455187\" class=\"hidden-answer\" style=\"display: none\">How many <i>x<\/i>\u2019s are there in [latex]x^{2}[\/latex]? That is, what is [latex]\\frac{{{x}^{2}}}{x}[\/latex]?<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image011.jpg\" alt=\"\" width=\"138\" height=\"21\" \/><\/p>\n<p>[latex]\\frac{{{x}^{2}}}{x}=x[\/latex]<i>. <\/i>Put <i>x<\/i> in the quotient above the [latex]-4x[\/latex]<i>\u00a0<\/i>term. (These are like terms, which helps to organize the problem.)<\/p>\n<p>Write the product of the divisor and the part of the quotient you just found under the dividend. Since [latex]x\\left(x+2\\right)=x^{2}+2x[\/latex],\u00a0write this underneath, and get ready to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image012.jpg\" alt=\"\" width=\"134\" height=\"58\" \/><\/p>\n<p>Rewrite [latex]\u2013\\left(x^{2} + 2x\\right)[\/latex]\u00a0as its opposite [latex]\u2013x^{2}\u20132x[\/latex]\u00a0so that you can add the opposite. (Adding the opposite is the same as subtracting, and it is easier to do.)<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image013.jpg\" alt=\"\" width=\"135\" height=\"55\" \/><\/p>\n<p>Add\u00a0[latex]-x^{2}[\/latex] to [latex]x^{2}[\/latex], and [latex]-2x[\/latex] to [latex]-4x[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064128\/image014.jpg\" alt=\"\" width=\"136\" height=\"69\" \/><\/p>\n<p>Bring down [latex]-12[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2253\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232007\/Screen-Shot-2016-03-28-at-4.19.50-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.19.50 PM\" width=\"152\" height=\"83\" \/><\/p>\n<p>Repeat the process. How many times does <i>x<\/i> go into [latex]-6x[\/latex]? In other words, what is [latex]\\frac{-6x}{x}[\/latex]?<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image015.jpg\" alt=\"\" width=\"126\" height=\"64\" \/><\/p>\n<p>Since [latex]\\frac{-6x}{x}=-6[\/latex], write [latex]-6[\/latex] in the quotient. Multiply [latex]-6[\/latex] and [latex]x+2[\/latex]\u00a0and prepare to subtract the product.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2254\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232512\/Screen-Shot-2016-03-28-at-4.24.52-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.24.52 PM\" width=\"165\" height=\"113\" \/><\/p>\n<p>Rewrite [latex]\u2013\\left(-6x\u201312\\right)[\/latex] as [latex]6x+12[\/latex], so that you can add the opposite.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image016.jpg\" alt=\"\" width=\"127\" height=\"86\" \/><\/p>\n<p>Add. In this case, there is no remainder, so you\u2019re done.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image017.jpg\" alt=\"\" width=\"134\" height=\"102\" \/><\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}=x\u20136[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Check this by multiplying:<\/p>\n<p style=\"text-align: center\">[latex]\\left(x-6\\right)\\left(x+2\\right)=x^{2}+2x-6x-12=x^{2}-4x-12[\/latex]<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>TRy It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3855\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3855&theme=oea&iframe_resize_id=ohm3855&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Polynomial long division involves many steps. \u00a0Hopefully this video will help you determine what step to do next when you are using polynomial long division.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1:  Divide a Trinomial by a Binomial Using Long Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KUPFg__Djzw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Let\u2019s try another example. In this example, a term is \u201cmissing\u201d from the dividend.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide: [latex]\\frac{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523374\">Show Solution<\/span><\/p>\n<div id=\"q523374\" class=\"hidden-answer\" style=\"display: none\">In setting up this problem, notice that there is an [latex]x^{3}[\/latex]\u00a0term but no [latex]x^{2}[\/latex]\u00a0term. Add [latex]0x^{2}[\/latex]\u00a0as a \u201cplace holder\u201d for this term. (Since 0 times anything is 0, you\u2019re not changing the value of the dividend.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image018.jpg\" alt=\"\" width=\"153\" height=\"18\" \/><\/p>\n<p>Focus on the first terms again: how many <i>x<\/i>\u2019s are there in [latex]x^{3}[\/latex]? Since [latex]\\frac{{{x}^{3}}}{x}=x^{2}[\/latex], put [latex]x^{2}[\/latex]\u00a0in the quotient.<\/p>\n<p>Multiply [latex]x^{2}\\left(x\u20133\\right)=x^{3}\u20133x^{2}[\/latex], write this underneath the dividend, and prepare to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064131\/image019.jpg\" alt=\"\" width=\"153\" height=\"49\" \/><\/p>\n<p>Rewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[\/latex]. Then add.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064132\/image020.jpg\" alt=\"\" width=\"153\" height=\"59\" \/><\/p>\n<p>Bring down the rest of the expression in the dividend. It\u2019s helpful to bring down <i>all<\/i> of the remaining terms.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064133\/image021.jpg\" alt=\"\" width=\"153\" height=\"59\" \/><\/p>\n<p>Now, repeat the process with the remaining expression, [latex]3x^{2}-6x\u201310[\/latex], as the dividend.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image022.jpg\" alt=\"\" width=\"153\" height=\"79\" \/><\/p>\n<p>Remember to watch the signs!<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image023.jpg\" alt=\"\" width=\"153\" height=\"89\" \/><\/p>\n<p>How many [latex]x[\/latex]\u2019s are there in [latex]3[\/latex]x? Since there are [latex]3[\/latex], multiply [latex]3\\left(x\u20133\\right)=3x\u20139[\/latex], write this underneath the dividend, and prepare to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064135\/image024.jpg\" alt=\"\" width=\"157\" height=\"118\" \/><\/p>\n<p>Continue until the <strong>degree<\/strong> of the remainder is <i>less <\/i>than the degree of the divisor. In this case the degree of the remainder, [latex]-1[\/latex], is [latex]0[\/latex], which is less than the degree of [latex]x-3[\/latex], which is [latex]1[\/latex].<\/p>\n<p>Also notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064221\/image025.jpg\" alt=\"\" width=\"153\" height=\"118\" \/><br \/>\nYou can write the remainder using the symbol R, or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\begin{array}{r}{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}=x^{2}+3x+3+R-1,\\\\x^{2}+3x+3+\\frac{-1}{x-3}, \\text{ or }\\\\x^{2}+3x+3-\\frac{1}{x-3}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Check the result:<\/p>\n<p style=\"text-align: center\">[latex]\\left(x\u20133\\right)\\left(x^{2}+3x+3\\right)\\,\\,\\,=\\,\\,\\,x\\left(x^{2}+3x+3\\right)\u20133\\left(x^{2}+3x+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}+3x^{2}+3x\u20133x^{2}\u20139x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,x^{3}\u20136x\u20139+\\left(-1\\right)\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u201310[\/latex]<\/p>\n<p>The video that follows shows another example of dividing a third degree trinomial by a first degree binomial.\u00a0 \u00a0Note the &#8220;missing term&#8221; and pay close attention to how we work with it.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Rxds7Q_UTeo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The process above works for dividing any polynomials no matter how many terms are in the divisor or the dividend and no matter what the degree of the polynomials are <em>as long as the degree of the dividend is greater than or equal to the degree of the divisor<\/em>. The main things to remember are:<\/p>\n<ul>\n<li>When subtracting, be sure to subtract the whole expression, not just the first term. <i>This is very easy to forget, so be careful!<\/i><\/li>\n<li>Stop when the degree of the remainder is less than the degree of the dividend or when you have brought down all the terms in the dividend and your quotient extends to the right edge of the dividend.<\/li>\n<\/ul>\n<p>The last video example shows how to divide a third degree trinomial by a second degree binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 6:  Divide a Polynomial by a Degree Two Binomial Using Long Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P6OTbUf8f60?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>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. \u00a0Dividing polynomials by polynomials of more than one term can be done using a process very much like long division of whole numbers. You must be careful to subtract entire expressions, not just the first term. Stop when the degree of the remainder is less than the degree of the divisor. The remainder can be written using R notation, or as a fraction added to the quotient with the remainder in the numerator and the divisor in the denominator.<\/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-16260\">\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\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169554,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Divide a Trinomial by a Binomial Using Long Division\",\"author\":\"James Sousa (Mathispower4u.com) 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