there is a mistake in the last example of this tutorial
it should be \( -\frac{4}{x} \)

<h2>Dividing a Polynomial By a Monomial</h2>
<p>In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a <strong>monomial</strong>.</p>
<p>The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.</p>
<p>\(\begin{array}{cccc}\phantom{\rule{4em}{0ex}}\text{The sum,}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{y}{5}+\frac{2}{5},\hfill \\ \phantom{\rule{4em}{0ex}}\text{simplifies to}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{y+2}{5}.\hfill \end{array}\)</p>
<p>Now we will do this in reverse to split a single fraction into separate fractions.</p>
<p>We’ll state the fraction addition property here just as you learned it and in reverse.</p>
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<h3>Fraction Addition</h3>
<p>If \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\) are numbers where \(c\ne 0\), then</p>
<p>\(\cfrac{a}{c}+\cfrac{b}{c}=\cfrac{a+b}{c}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\cfrac{a+b}{c}=\cfrac{a}{c}+\cfrac{b}{c}\)</p>
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<p>We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.</p>
<p>\(\begin{array}{cccc}\phantom{\rule{4em}{0ex}}\text{For example,}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{y+2}{5}\hfill \\ \phantom{\rule{4em}{0ex}}\text{can be written}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{y}{5}+\frac{2}{5}.\hfill \end{array}\)</p>
<p>We use this form of fraction addition to divide polynomials by monomials.</p>
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<h3>Division of a Polynomial by a Monomial</h3>
<p>To divide a polynomial by a monomial, divide each term of the polynomial by the <strong>monomial</strong>.</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\frac{7{y}^{2}+21}{7}.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \hfill \frac{7{y}^{2}+21}{7}\hfill \\ \text{Divide each term of the numerator by the denominator.}\hfill &amp; &amp; &amp; \hfill \frac{7{y}^{2}}{7}+\frac{21}{7}\hfill \\ \text{Simplify each fraction.}\hfill &amp; &amp; &amp; \hfill {y}^{2}+3\hfill \end{array}\)</p>
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<p>Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\left(18{x}^{3}-36{x}^{2}\right)÷6x.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \left(18{x}^{3}-36{x}^{2}\right)÷6x\hfill \\ \text{Rewrite as a fraction.}\hfill &amp; &amp; &amp; \frac{18{x}^{3}-36{x}^{2}}{6x}\hfill \\ \text{Divide each term of the numerator by the denominator.}\hfill &amp; &amp; &amp; \frac{18{x}^{3}}{6x}-\frac{36{x}^{2}}{6x}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; 3{x}^{2}-6x\hfill \end{array}\)</p>
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<p>When we divide by a negative, we must be extra careful with the signs.</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\frac{12{d}^{2}-16d}{-4}.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \hfill \frac{12{d}^{2}-16d}{-4}\hfill \\ \text{Divide each term of the numerator by the denominator.}\hfill &amp; &amp; &amp; \hfill \frac{12{d}^{2}}{-4}-\frac{16d}{-4}\hfill \\ \text{Simplify. Remember, subtracting a negative is like adding a positive!}\hfill &amp; &amp; &amp; \hfill -3{d}^{2}+4d\hfill \end{array}\)</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \hfill \phantom{\rule{4em}{0ex}}\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}\hfill \\ \text{Separate the terms.}\hfill &amp; &amp; &amp; \hfill \phantom{\rule{4em}{0ex}}\frac{105{y}^{5}}{5{y}^{2}}+\frac{75{y}^{3}}{5{y}^{2}}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \hfill \phantom{\rule{4em}{0ex}}21{y}^{3}+15y\hfill \end{array}\)</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\left(15{x}^{3}y-35x{y}^{2}\right)÷\left(-5xy\right).\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \left(15{x}^{3}y-35x{y}^{2}\right)÷\left(-5xy\right)\hfill \\ \text{Rewrite as a fraction.}\hfill &amp; &amp; &amp; \frac{15{x}^{3}y-35x{y}^{2}}{-5xy}\hfill \\ \text{Separate the terms. Be careful with the signs!}\hfill &amp; &amp; &amp; \frac{15{x}^{3}y}{-5xy}-\frac{35x{y}^{2}}{-5xy}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; -3{x}^{2}+7y\hfill \end{array}\)</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\frac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}\hfill \\ \text{Separate the terms.}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{36{x}^{3}{y}^{2}}{9{x}^{2}y}+\frac{27{x}^{2}{y}^{2}}{9{x}^{2}y}-\frac{9{x}^{2}{y}^{3}}{9{x}^{2}y}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{5.7em}{0ex}}4xy+3y-{y}^{2}\hfill \end{array}\)</p>
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<h2>Example</h2>
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<p>Find the quotient: \(\frac{10{x}^{2}+5x-20}{5x}.\)</p>
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<h3>Solution</h3>
<p>\(\begin{array}{cccc}&amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{10{x}^{2}+5x-20}{5x}\hfill \\ \text{Separate the terms.}\hfill &amp; &amp; &amp; \phantom{\rule{4em}{0ex}}\frac{10{x}^{2}}{5x}+\frac{5x}{5x}-\frac{20}{5x}\hfill \\ \text{Simplify.}\hfill &amp; &amp; &amp; \phantom{\rule{5.2em}{0ex}}2x+1+\frac{4}{x}\hfill \end{array}\)</p>
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Dividing a Polynomial By a Monomial

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

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Discover how to identify, evaluate, add, subtract, multiply, and divide various types of polynomials, and simplify expressions using exponents and scientific notations in this tutorial.


Dividing a Polynomial By a Monomial

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