# 1.2.5: Dividing Polynomials

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##### Learning Objectives

By the end of this section, you will be able to:

• Reduce fractions
• Divide a monomial by a monomial
• Divide a polynomial by a monomial
• Divide polynomials using long division
##### Be Prepared

Before you get started, take this readiness quiz.

1. Find the greatest common factor (GCF) of $$35$$ and $$49$$.

2. Reduce $$\dfrac{35}{49}$$.

## Reducing Fractions

Let's say we want to reduce the fraction $$\dfrac{60}{84}$$. We need to find the largest integer that goes into $$60$$ and $$84$$ at the same time. Note that $$3$$ goes into $$60$$ and $$84$$ since

$$60=3\cdot 20\qquad\text{and}\qquad 84 = 3\cdot 28.$$

So $$3$$ is a common factor. However, $$4$$ goes into $$20$$ and $$28$$ and so we can write

$$60=3\cdot (4\cdot 5)\qquad\text{and}\qquad 84 = 3\cdot (4\cdot 7)$$

or

$$60=(3\cdot 4)\cdot 5\qquad\text{and}\qquad 84 = (3\cdot 4)\cdot 7.$$

This means that $$3\cdot 4=12$$ goes into $$60$$ and $$84$$ and we can write:

$$60=12\cdot 5\qquad\text{and}\qquad 84 = 12\cdot 7.$$

So $$12$$ is also a common factor of $$60$$ and $$84$$. Because $$5$$ and $$12$$ do not have any common factor other than $$1$$, the greatest common factor of $$60$$ and $$84$$ is $$12$$. Now we use the cancelation property:

$$\dfrac{a\cdot b}{a\cdot c}=\dfrac{b}{c}$$

to simplify $$\dfrac{60}{84}$$:

\begin{align*}\dfrac{60}{84} &= \dfrac{12\cdot 5}{12\cdot 7}\\ &=\dfrac{5}{7}.\end{align*}

The reduction is $$\dfrac{5}{7}$$. We want to keep the fraction format. The two fractions $$\dfrac{60}{84}$$ and  $$\dfrac{5}{7}$$ are said to be equivalent as they represent the same number. If we had used the common factor $$3$$ that we found in the beginning, we would have gotten

\begin{align*}\dfrac{60}{84} &= \dfrac{3\cdot 20}{3\cdot 28}\\ &=\dfrac{20}{28}\end{align*}

so that  $$\dfrac{60}{84}$$,  $$\dfrac{5}{7}$$  and $$\dfrac{20}{28}$$ are all equivalent, but  $$\dfrac{5}{7}$$ is the one that cannot be simplified any further. This is the reduction we want.

##### Example $$\PageIndex{1}$$

Simplify:

a. $$\dfrac{36}{21}$$

b. $$\dfrac{72}{44}$$

c. $$\dfrac{8}{64}$$

###### Solution

a. The greatest common factor of $$36$$ and $$21$$ is $$3$$.

\begin{align*} \dfrac{36}{21} & = \dfrac{3\cdot 12}{3\cdot 7}\\ & = \dfrac{12}{7}\end{align*}

b. The greatest common factor of $$72$$ and $$44$$ is $$4$$.

\begin{align*} \dfrac{72}{44} & = \dfrac{4\cdot 18}{4\cdot 11}\\ & = \dfrac{18}{11}\end{align*}

c. The greatest common factor of $$8$$ and $$64$$ is $$8$$.

\begin{align*} \dfrac{8}{64} & = \dfrac{8\cdot 1}{8\cdot 8}\\ & = \dfrac{1}{8}\end{align*}

##### Try It $$\PageIndex{2}$$

Simplify:

a. $$\dfrac{108}{16}$$

b. $$\dfrac{25}{100}$$

c. $$\dfrac{81}{3}$$

a. $$\dfrac{27}{4}$$

b. $$\dfrac{1}{4}$$

c. $$27$$

## Dividing Monomials

Now we will look at some examples where dividing two monomials results in a monomial (which is not always the case!!).

Consider $$\quad\dfrac{x^5}{x^2}$$
What do they mean? $$=\dfrac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}$$
Use the Equivalent Fractions Property. $$=\dfrac{\cancel{x}\cdot\cancel{x}\cdot x\cdot x\cdot x}{\cancel{x}\cdot \cancel{x}}$$
Simplify. $$=x^3$$
Note.

$$\quad\dfrac{x^5}{x^2}=\dfrac{x^2x^3}{x^2}=x^3$$

Reducing the fraction.

##### Example $$\PageIndex{3}$$

Find the quotient $$54a^2b^3÷ (−6ab^2)$$.

###### Solution

When we divide monomials with more than one variable, we write one fraction for each variable.

$$\quad 54a^2b^3÷(−6ab^2)$$
Rewrite as a fraction. $$=\dfrac{54a^2b^3}{−6ab^2}$$
Use fraction multiplication. $$=\dfrac{54}{−6}\cdot\dfrac{a^2}{a}\cdot\dfrac{b^3}{b^2}$$
Reduce the fraction. $$=−9ab$$

### Try It $$\PageIndex{4}$$

Find the quotient $$−72a^7b^3÷(8a^{5}b^2)$$.

$$−{9}{a^2b}$$

### Try It $$\PageIndex{5}$$

Find the quotient $$−63c^8d^3÷(7c^{2}d)$$.

$$−9c^6d^2$$

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

##### Example $$\PageIndex{6}$$

Find the quotient $$\dfrac{14x^7y^{12}}{21x^{3}y^6}$$.

###### Solution

Be very careful to simplify $$\dfrac{14}{21}$$ by dividing out a common factor, and to simplify the variables by subtracting their exponents.

$$\quad\dfrac{14x^7y^{12}}{21x^{3}y^6}$$
Use fraction multiplication. $$=\dfrac{14}{21}\cdot \dfrac{x^7}{x^3}\cdot\dfrac{y^{12}}{y^6}$$
Reduce. $$=\dfrac{2}{3}x^4y^6$$

## Dividing a Polynomial by a Monomial

Do example with numbers here for distribution of division

Now that we know how to divide a monomial by a monomial, 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.

The sum $$\dfrac{3}{7}+\dfrac{2}{7}=\dfrac{3+2}{7}$$. So it is also true that $$\dfrac{3+2}{7}= \dfrac{3}{7}+\dfrac{2}{7}$$, so you can distribute the divion by 7. You may also recall that division by 7 is the same as multiplication by $$\dfrac{1}{7}$$ so, $$\dfrac{3+2}{7}=\dfrac{1}{7}(3+2)=\dfrac{1}{7}\cdot 3+\dfrac{1}{7}\cdot 2 =\dfrac{3}{7}+\dfrac{2}{7}$$, or in words because you distribute multiplication over addition and subtraction, and division is multiplication is division by a reciprocal, you can also distribute division over addition and subtraction.

Here is another example with a variable.

The sum $$\dfrac{y}{5}+\dfrac{2}{5}$$ simplifies to $$\dfrac{y+2}{5}$$. Now we will do this in reverse to split a single fraction into separate fractions. For example, $$\dfrac{y+2}{5}$$ can be written $$\dfrac{y}{5}+\dfrac{2}{5}$$.  Or, thinking of this as distibution: $$\dfrac{y+2}{5}=(y+2)\cdot\dfrac15=\dfrac{y}{5}+\dfrac25.$$

This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where $$c\neq 0$$, then $$\dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}$$. We will use this 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, or in other words, we distribute the division over addition and subtraction.

##### Example $$\PageIndex{7}$$

Find the quotient $$(18x^3y−36xy^2)÷(−3xy)$$.

###### Solution
 $$\quad (18x^3y−36xy^2)÷(−3xy)$$ Rewrite as a fraction. $$=\dfrac{18x^3y−36xy^2}{−3xy}$$ Divide each term by the divisor (distribute the division). Be careful with the signs! $$=\dfrac{18x^3y}{−3xy}−\dfrac{36xy^2}{−3xy}$$ Simplify. $$=−6x^2+12y$$

### Try It $$\PageIndex{8}$$

Find the quotient $$(32a^2b−16ab^2)÷(−8ab)$$.

$$−4a+2b$$

### Try It $$\PageIndex{9}$$

Find the quotient $$(−48a^8b^4−36a^6b^5)÷(−6a^3b^3)$$.

$$8a^5b+6a^3b^2$$

We may, in certain situations, also divide a polynomial by a binomial as in the following example.

##### Example $$\PageIndex{10}$$

Find the quotient $$(6(x-2)(3x-2))÷(3(x-2))$$.

###### Solution
 $$\quad (6(x-2)(3x-2))÷(3(x-2))$$ Rewrite as a fraction. $$=\dfrac{6(x-2))(3x-2)}{3(x-2)}$$ Identify the common factors. $$=\dfrac{3(x-2)2(3x-2)}{3(x-2)}$$ Simplify. $$=2(3x-2)$$ or $$6x-4$$
##### Try It $$\PageIndex{11}$$

Find the quotient $$((-4(2x-1))(2x+7))÷2(2x+7)$$.

$$-2(2x-1)$$

##### Try It $$\PageIndex{12}$$

Find the quotient $$((4(2x-1))(2x+7))÷-4(2x-1)$$.

$$-(2x+7)$$

## Dividing Polynomials Using Long Division

Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

$\begin{array}{r} 35\phantom{5}\\ 25{\overline{\smash{\big)}\,875}}\\ \underline{-750}\\125\\\underline{-125}\\ 0\nonumber\end{array}$

The quotient is $$35$$ and the remainder is $$0$$. We check division by multiplying the quotient by the divisor and then adding the remainder. If we did the division correctly, the result should equal the dividend.

$35·25+0=875\checkmark\nonumber$

Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

##### Example $$\PageIndex{13}$$

Find the quotient and the remainder of $$(x^2+9x+20)÷(x+5)$$.

###### Solution
 $$(x^2+9x+20)÷(x+5)$$ Write it as a long division problem. Be sure the dividend is in standard form. \begin{array}{r}  x+5{\overline{\smash{\big)}\,x^2+9x+20\phantom{)}}}\\ \nonumber\end{array} Divide $$x^2$$ by $$x$$. It may help to ask yourself, “What do I need to multiply $$x$$ by to get $$x^2$$?” Put the answer, $$x$$, in the quotient over the $$x$$ term. \begin{array}{r} x\phantom{x+20)}\\ x+5{\overline{\smash{\big)}\,x^2+9x+20\phantom{)}}}\\ \nonumber\end{array} Multiply $$x$$ by $$x+5$$. Change the sign of each term and put the answer under $$x^2+9x$$. \begin{array}{r} x\phantom{(x+20)}\\ x+5{\overline{\smash{\big)}\,x^2+9x+20\phantom{)}}}\\ \underline{-x^2-5x\phantom{\;+20)}}\nonumber\end{array} Add it to $$x^2+9x$$. \begin{array}{r} x\phantom{x+20)}\\ x+5{\overline{\smash{\big)}\,x^2+9x+20\phantom{)}}}\\ \underline{-x^2-5x\phantom{\;+20)}} \\ 4x+20\phantom{)}\\ \nonumber\end{array} Divide $$4x$$ by $$x$$. It may help to ask yourself, “What do I need to multiply $$x$$ by to get $$4x$$?” Put the answer, $$4$$, in the quotient over the constant term. \begin{array}{r} x+4\phantom{20)}\\ x+5{\overline{\smash{\big)}\,x^2+9x+20\phantom{)}}}\\ \underline{-x^2-5x\phantom{\;+20)}} \\4x+20\phantom{)}\\ \nonumber\end{array} Multiply 4 by $$x+5$$. Change the sign of each term and put the answer under $$4x+20$$. \begin{array}{r} x+4\phantom{20)}\\ x+5{\overline{\smash{\big)}\,x^2+9x+20\phantom{)}}}\\ \underline{-x^2-5x\phantom{\;+20)}} \\4x+20\phantom{)}\\ \underline{~\phantom{()}-4x-20} \nonumber\end{array} Add it to $$4x+20$$. \begin{array}{r} x+4\phantom{)}\\ x+5{\overline{\smash{\big)}\phantom{-} x^2+9x+20\phantom{)}}}\\ \underline{~\phantom{(}-x^2+(-5x)\phantom{\;+20)}}\\ 4x+20\phantom{)}\\ \underline{~\phantom{()}-4x-20}\\ 0 \phantom{)}\nonumber\end{array} Check. $$\begin{array} {ll} {\text{Multiply the quotient by the divisor.}} &{(x+4)(x+5)} \\ {\text{You should get the dividend.}} &{x^2+9x+20\checkmark}\\ \end{array}$$ Conclude. The quotient of $$(x^2+9x+20)÷(x+5)$$ is $$x+4$$, and the remainder is $$0$$.

##### Try It $$\PageIndex{14}$$

Find the quotient and the remainder of $$(y^2+10y+21)÷(y+3)$$.

The quotient is $$y+7$$. The remainder is $$0$$.

##### Try It $$\PageIndex{15}$$

Find the quotient and the remainder of $$(m^2+9m+20)÷(m+4)$$.

The quotient is $$m+5$$. The remainder is $$0$$.

Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be $$x^4−x^2+5x−6$$. It is missing an $$x^3$$ term. We will add in $$0x^3$$ as a placeholder.

##### Example $$\PageIndex{16}$$

Find the quotient and the remainder of $$(x^4−x^2+5x−6)÷(x+2)$$.

###### Solution

Notice that there is no $$x^3$$ term in the dividend. We will add $$0x^3$$ as a placeholder.

$$(x^4−x^2+5x−6)÷(x+2)$$
Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms. \begin{array}{r}  x+2{\overline{\smash{\big)}\phantom{-} x^4+0x^3-x^2+5x-2\phantom{)}}}\nonumber\end{array}
Divide $$x^4$$ by $$x$$.
Put the answer, $$x^3$$, in the quotient over the $$x^3$$ term.
Multiply $$x^3$$ by $$x+2$$. Change the sign of each term and put the answer under $$x^4+0x^3$$. Line up the like terms. Add them.

\begin{array}{r} x^3\phantom{)-2x^2+3x-1)}\\ x+2{\overline{\smash{\big)}\phantom{-} x^4+0x^3-x^2+5x-2\phantom{)}}}\\  \underline{ -x^4-2x^3\phantom{\;-x^2+5x-2)}}\\ -2x^3-x^2+5x-2 \phantom{\;)}\nonumber\end{array}

Divide $$−2x^3$$ by $$x$$.
Put the answer, $$−2x^2$$, in the quotient over the $$x^2$$ term.
Multiply $$−2x^2$$ times $$x+1$$. Change the sign of each term and put the answer under Line up the like terms. Add them.

\begin{array}{r} x^3-2x^2\phantom{))}\\ x+2{\overline{\smash{\big)}\phantom{-} x^4+0x^3-x^2+5x-2\phantom{)}}}\\  \underline{ -x^4-2x^3\phantom{\;-x^2+5x-2)}}\\ -2x^3-x^2+5x-2 \phantom{\;)}\\  \underline{2x^3+4x^2\phantom{\;+5x-2}}\\ 3x^2+5x-2  \phantom{\;)} \\ \nonumber\end{array}

Divide $$3x^2$$ by $$x$$.
Put the answer, $$3x$$, in the quotient over the $$x$$ term.
Multiply $$3x$$ times $$x+1$$. Line up the like terms.
Subtract and bring down the next term.

\begin{array}{r} x^3-2x^2+3x\phantom{))}\\ x+2{\overline{\smash{\big)}\phantom{-} x^4+0x^3-x^2+5x-2\phantom{)}}}\\  \underline{ -x^4-2x^3\phantom{\;-x^2+5x-2)}}\\ -2x^3-x^2+5x-2 \phantom{\;)}\\  \underline{2x^3+4x^2\phantom{\;+5x-2}}\\ 3x^2+5x-2  \phantom{\;)} \\ \underline{~-3x^2-6x\phantom{)\;-2}}\\ -x-2\phantom{\;)} \\ \nonumber\end{array}

Divide $$−x$$ by $$x$$.
Put the answer, $$−1$$, in the quotient over the constant term.
Multiply $$−1$$ times $$x+1$$. Line up the like terms.
Change the signs, add.

\begin{array}{r} x^3-2x^2+3x-1\phantom{))}\\ x+2{\overline{\smash{\big)}\phantom{-} x^4+0x^3-x^2+5x-2\phantom{)}}}\\  \underline{ -x^4-2x^3\phantom{\;-x^2+5x-2)}}\\ -2x^3-x^2+5x-2 \phantom{\;)}\\  \underline{2x^3+4x^2\phantom{\;+5x-2}}\\ 3x^2+5x-2  \phantom{\;)} \\ \underline{~-3x^2-6x\phantom{)\;-2}}\\ -x-2\phantom{\;)} \\ \underline{~x+2}\\ 0\phantom{\;)}\nonumber\end{array}

Check. Multiply $$(x+2)(x^3−2x^2+3x−1−4x+2)$$.
The result should be $$x^4−x^2+5x−6$$.
Conclude. The quotient of $$(x^4−x^2+5x−6)÷(x+2)$$ is $$x^3-2x^2+3x-1$$, and the remainder is $$0$$.
##### Try It $$\PageIndex{17}$$

Find the quotient and the remainder of $$(x^4−7x^2+7x+6)÷(x+3)$$.

The quotient is $$x^3−3x^2+2x+1+3x+3$$. The remainder is $$0$$.

##### Try It $$\PageIndex{18}$$

Find the quotient and the remainder of $$(x^4−11x^2−7x−6)÷(x+3)$$.

The quotient is $$x^3−3x^2−2x−1−3x+3$$. The remainder is $$0$$.

In the next example, we will divide by $$2a−3$$. As we divide, we will have to consider the constants as well as the variables.

##### Example $$\PageIndex{19}$$

Find the quotient and the remainder of $$(8a^3+27)÷(2a+3)$$.

###### Solution

This time we will show the division all in one step. We need to add two placeholders in order to divide.

\begin{array}{r} 4a^2-6a+9\phantom{))}\\ 2a+3{\overline{\smash{\big)}\phantom{-} 8a^3+0a^2+0a+27\phantom{)}}}\\ \underline{~ -8a^3-12a^2\phantom{\;+0a+27)}}\\ -12a^2+0a+27 \phantom{\;)}\\ \underline{12a^2+18a\phantom{\;+27}}\\ 18a+27  \phantom{\;)} \\ \underline{~-18a-27\phantom{)}}\\ 0\phantom{\;)} \nonumber\end{array}

To check, multiply $$(2a+3)(4a^2−6a+9)$$. The result should be $$8a^3+27$$.

The quotient of $$(8a^3+27)÷(2a+3)$$ is $$4a^2-6a+9$$ and the remainder is $$0$$.

###### Try It $$\PageIndex{20}$$

Find the quotient and the remainder of $$(x^3−64)÷(x−4)$$.

The quotient is $$x^2+4x+16$$. The remainder is $$0$$.

###### Try It $$\PageIndex{21}$$

Find the quotient: $$(125x^3−8)÷(5x−2)$$.

The quotient is $$25x^2+10x+4$$. The remainder is $$0$$.

When we divided 875 by 25, we had remainder zero. But sometimes division of numbers does leave a remainder different from zero. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder that is not zero. The degree of the remainder is always less than the degree of the divisor. To check, we need to verify that $$(\text{quotient})(\text{divisor}) + \text{remainder} = \text{dividend}$$.

##### Example $$\PageIndex{22}$$

Find the quotient and remainder when $$2x^3+3x^2+x+8$$ is divided by $$x+2$$.

###### Solution
 $$(2x^3+3x^2+x+8)÷(x+2)$$ Add it to $$4x+20$$. \begin{array}{r} 2x^2-x+3\phantom{))}\\ x+2{\overline{\smash{\big)}\phantom{-} 2x^3+3x^2+x+8\phantom{)}}}\\ \underline{~\phantom{(}-2x^3-4x^2\phantom{\;+x+8)}}\\ -x^2+x+8 \phantom{\;)}\\ \underline{x^2+2x\phantom{+8}}\\ 3x+8  \phantom{\;)} \\ \underline{~\phantom{(}-3x-6\phantom{)}}\\ 2\phantom{\;)} \nonumber\end{array} Check. \begin{align} (\text{quotient})(\text{divisor}) + \text{remainder} &= \text{dividend} \nonumber\\ (2x^2−1x+3)(x+2)+2 &\overset{?}{=} 2x^3+3x^2+x+8 \nonumber\\ 2x^3−x^2+3x+4x^2−2x+6+2 &\overset{?}{=} 2x^3+3x^2+x+8 \nonumber\\ 2x^3+3x^2+x+8 &= 2x^3+3x^2+x+8\checkmark \nonumber \end{align} Conclude. The quotient of $$(2x^3+3x^2+x+8)÷(x+2)$$ is $$2x^2-x+3$$, and the remainder is $$2$$.
###### Try It $$\PageIndex{23}$$

Find the quotient and remainder when $$3x^3+10x^2+6x−2$$ is divided by $$x+2$$.

The quotient is $$3x^2+4x−2$$. The remainder is $$2$$.

###### Example $$\PageIndex{24}$$

Find the quotient and remainder when $$4x^3+5x^2−5x+3$$ is divided by $$x+2$$.

The quotient is $$4x^2−3x+1$$. The remainder is $$1$$.

##### Writing Exercises $$\PageIndex{25}$$
1. Explain why you can distribute division over addition and subtraction.
2. Can you divide a polynomial by a monomial and not get a polynomial?  Give an example.
3. Give another example of the type in the last exercise.
4. What is the first step to reducing a fraction (numerator and denominator whole numbers)? Explain.
##### Exit Problem
1. Divide $$(27x^3y^7-18x^7y^3+9x^2y^3)\div (9x^2y^3)$$.
2. Divide $$x^3-8x+3$$ by $$x+5$$ using long division.

## Key Concepts

• Dividend, divisor, quotient, remainder
• Division of a polynomial by a monomial
• Long division of polynomials

1.2.5: Dividing Polynomials is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.