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Dividing a Polynomial By a Monomial

Dividing a Polynomial By a Monomial

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 monomial.

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.

\(\begin{array}{cccc}\phantom{\rule{4em}{0ex}}\text{The sum,}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{y}{5}+\frac{2}{5},\hfill \\ \phantom{\rule{4em}{0ex}}\text{simplifies to}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{y+2}{5}.\hfill \end{array}\)

Now we will do this in reverse to split a single fraction into separate fractions.

We’ll state the fraction addition property here just as you learned it and in reverse.

Fraction Addition

If \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\) are numbers where \(c\ne 0\), then

\(\cfrac{a}{c}+\cfrac{b}{c}=\cfrac{a+b}{c}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\cfrac{a+b}{c}=\cfrac{a}{c}+\cfrac{b}{c}\)

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.

\(\begin{array}{cccc}\phantom{\rule{4em}{0ex}}\text{For example,}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{y+2}{5}\hfill \\ \phantom{\rule{4em}{0ex}}\text{can be written}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{y}{5}+\frac{2}{5}.\hfill \end{array}\)

We use this form of fraction addition to divide polynomials by monomials.

Division of a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Example

Find the quotient: \(\frac{7{y}^{2}+21}{7}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \frac{7{y}^{2}+21}{7}\hfill \\ \text{Divide each term of the numerator by the denominator.}\hfill & & & \hfill \frac{7{y}^{2}}{7}+\frac{21}{7}\hfill \\ \text{Simplify each fraction.}\hfill & & & \hfill {y}^{2}+3\hfill \end{array}\)

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.

Example

Find the quotient: \(\left(18{x}^{3}-36{x}^{2}\right)÷6x.\)

Solution

\(\begin{array}{cccc}& & & \left(18{x}^{3}-36{x}^{2}\right)÷6x\hfill \\ \text{Rewrite as a fraction.}\hfill & & & \frac{18{x}^{3}-36{x}^{2}}{6x}\hfill \\ \text{Divide each term of the numerator by the denominator.}\hfill & & & \frac{18{x}^{3}}{6x}-\frac{36{x}^{2}}{6x}\hfill \\ \text{Simplify.}\hfill & & & 3{x}^{2}-6x\hfill \end{array}\)

When we divide by a negative, we must be extra careful with the signs.

Example

Find the quotient: \(\frac{12{d}^{2}-16d}{-4}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \frac{12{d}^{2}-16d}{-4}\hfill \\ \text{Divide each term of the numerator by the denominator.}\hfill & & & \hfill \frac{12{d}^{2}}{-4}-\frac{16d}{-4}\hfill \\ \text{Simplify. Remember, subtracting a negative is like adding a positive!}\hfill & & & \hfill -3{d}^{2}+4d\hfill \end{array}\)

Example

Find the quotient: \(\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{105{y}^{5}+75{y}^{3}}{5{y}^{2}}\hfill \\ \text{Separate the terms.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{105{y}^{5}}{5{y}^{2}}+\frac{75{y}^{3}}{5{y}^{2}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}21{y}^{3}+15y\hfill \end{array}\)

Example

Find the quotient: \(\left(15{x}^{3}y-35x{y}^{2}\right)÷\left(-5xy\right).\)

Solution

\(\begin{array}{cccc}& & & \left(15{x}^{3}y-35x{y}^{2}\right)÷\left(-5xy\right)\hfill \\ \text{Rewrite as a fraction.}\hfill & & & \frac{15{x}^{3}y-35x{y}^{2}}{-5xy}\hfill \\ \text{Separate the terms. Be careful with the signs!}\hfill & & & \frac{15{x}^{3}y}{-5xy}-\frac{35x{y}^{2}}{-5xy}\hfill \\ \text{Simplify.}\hfill & & & -3{x}^{2}+7y\hfill \end{array}\)

Example

Find the quotient: \(\frac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}.\)

Solution

\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\frac{36{x}^{3}{y}^{2}+27{x}^{2}{y}^{2}-9{x}^{2}{y}^{3}}{9{x}^{2}y}\hfill \\ \text{Separate the terms.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{36{x}^{3}{y}^{2}}{9{x}^{2}y}+\frac{27{x}^{2}{y}^{2}}{9{x}^{2}y}-\frac{9{x}^{2}{y}^{3}}{9{x}^{2}y}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{5.7em}{0ex}}4xy+3y-{y}^{2}\hfill \end{array}\)

Example

Find the quotient: \(\frac{10{x}^{2}+5x-20}{5x}.\)

Solution

\(\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}\frac{10{x}^{2}+5x-20}{5x}\hfill \\ \text{Separate the terms.}\hfill & & & \phantom{\rule{4em}{0ex}}\frac{10{x}^{2}}{5x}+\frac{5x}{5x}-\frac{20}{5x}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{5.2em}{0ex}}2x+1+\frac{4}{x}\hfill \end{array}\)

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