Dividing Polynomials

Dividing Monomials

We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.

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.

Divide a Polynomial by a Monomial

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 $\frac{y}{5}+\frac{2}{5}$

simplifies to $\frac{y+2}{5}.$

Now we will do this in reverse to split a single fraction into separate fractions. For example, $\frac{y+2}{5}$

can be written $\frac{y}{5}+\frac{2}{5}.$

This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where $c\ne 0,$

then $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}.$

We will use this to divide polynomials by monomials.

Divide 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.

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

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.

Find the quotient: $\left(x^2+9x+20\right)÷\left(x+5\right).$

Write it as a long division problem. Be sure the dividend is in standard form.
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. Multiply $x$ times $x+5.$ Line up the like terms under the dividend.
Subtract $x^2+5x$ from $x^2+9x.$ You may find it easier to change the signs and then add. Then bring down the last term, 20.
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.
Multiply 4 times $x+5.$
Subtract $4x+20$ from $4x+20.$
Check: Multiply the quotient by the divisor. $\left(x+4\right)\left(x+5\right)$ You should get the dividend. $x^2+9x+20✓$

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

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.

Find the quotient: $\left(x^4-x^2+5x-6\right)÷\left(x+2\right).$

Notice that there is no $x^3$

term in the dividend. We will add $0x^3$

as a placeholder.

Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.
Divide $x^4$ by $x.$ Put the answer, $x^3,$ in the quotient over the $x^3$ term. Multiply $x^3$ times $x+2.$ Line up the like terms. Subtract and then bring down the next term.
Divide $\mathrm{-2}{x}^3$ by $x.$ Put the answer, $\mathrm{-2}{x}^2,$ in the quotient over the $x^2$ term. Multiply $\mathrm{-2}{x}^2$ times $x+1.$ Line up the like terms Subtract and bring down the next term.
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.
Divide $\text{−}x$ by $x.$ Put the answer, $\mathrm{-1},$ in the quotient over the constant term. Multiply $\mathrm{-1}$ times $x+1.$ Line up the like terms. Change the signs, add. Write the remainder as a fraction with the divisor as the denominator.
To check, multiply $\left(x+2\right)\left(x^3-2x^2+3x-1-\frac{4}{x+2}\right)$. The result should be $x^4-x^2+5x-6.$

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.

Divide Polynomials using Synthetic Division

As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in [link] and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.

Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the $x$

and $x^2$

are removed. as well as the $\text{−}{x}^2$

and $\mathrm{-4}x$

as they are opposite the term above.

The first row of the synthetic division is the coefficients of the dividend. The $\mathrm{-5}$

is the opposite of the 5 in the divisor.

The second row of the synthetic division are the numbers shown in red in the division problem.

The third row of the synthetic division are the numbers shown in blue in the division problem.

Notice the quotient and remainder are shown in the third row.

The following example will explain the process.

Use synthetic division to find the quotient and remainder when $2x^3+3x^2+x+8$

is divided by $x+2.$

Write the dividend with decreasing powers of $x.$
Write the coefficients of the terms as the first row of the synthetic division.
Write the divisor as $x-c$ and place c in the synthetic division in the divisor box.
Bring down the first coefficient to the third row.
Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
Add the second column, putting the result in the third row.
Multiply that result by the divisor and place the result in the second row under the third coefficient.
Add the third column, putting the result in the third row.
Multiply that result by the divisor and place the result in the third row under the third coefficient.
Add the final column, putting the result in the third row.
The quotient is $2x^2-1x+3$ and the remainder is 2.

The division is complete. The numbers in the third row give us the result. The $2\mathrm{-1}3$

are the coefficients of the quotient. The quotient is $2x^2-1x+3.$

The 2 in the box in the third row is the remainder.

Check:

$\begin{array}{}\\ \\ \hfill \text{(quotient)(divisor)}+\text{remainder}& =\hfill & \text{dividend}\hfill \\ \hfill \left(2x^2-1x+3\right)\left(x+2\right)+2& \stackrel{?}{=}\hfill & 2x^3+3x^2+x+8\hfill \\ \hfill 2x^3-x^2+3x+4x^2-2x+6+2& \stackrel{?}{=}\hfill & 2x^3+3x^2+x+8\hfill \\ \hfill 2x^3+3x^2+x+8& =\hfill & 2x^3+3x^2+x+8✓\hfill \end{array}$

In the next example, we will do all the steps together.

Divide Polynomial Functions

Just as polynomials can be divided, polynomial functions can also be divided.

Use the Remainder and Factor Theorem

Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as $x-c,$

the value of the function at $c,f\left(c\right),$

is the same as the remainder from the division problem.

| 3 | {: valign=”top”}{: summary=”The table has four rows and five columns. The first row is a header row with the headers “dividend”, “Divisor”, “Remainder”, and f of c. The second row contains the expressions x to the fourth power minus x squared plus 5 x minus 6, x minus negative 2, negative 4, f of x equals x to the fourth power minus x squared plus 5 x minus 6, negative 4. The third row contains the expressions 3 x cubed minus 2 x squared minus 10 x plus 8, x minus 2, 4, f of x equals 3 x cubed minus 2 x squared minus 10 x plus 8, 4. The fourth row contains the expressions x to the fourth power minus 16 x squared plus 3 x plus 15, x minus negative 4, 3, f of x equals x to the fourth minus 16 x squared plus 3 x plus 15, 3.”}

To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend $f\left(x\right),$

we multiply the quotient, $q\left(x\right)$

times the divisor, $x-c,$

and add the remainder, r.

{: valign=”top”}	If we evaluate this at $c,$

we get: | | {: valign=”top”}| | | {: valign=”top”}| | | {: valign=”top”}{: .unnumbered .unstyled summary=”The figure shows four equations in function notation. The first equation is f of x equals q of x times the quantity x minus c in parentheses plus r. The second equation replaces x with c. The second equation is f of c equals q of c times the quantity c minus c in parentheses plus r. The third equation simplifies the difference c minus c. The third equation is f of c equals q of c times 0 plus r. The fourth equation simplifies the product q of c times 0. The fourth equation is f of c equals r.” data-label=””}

This leads us to the Remainder Theorem.

Use the Remainder Theorem to find the remainder when $f\left(x\right)=x^3+3x+19$

is divided by $x+2.$

To use the Remainder Theorem, we must use the divisor in the $x-c$

form. We can write the divisor $x+2$

as $x-\left(\mathrm{-2}\right).$

So, our $c$

is $\mathrm{-2}.$

To find the remainder, we evaluate $f\left(c\right)$

which is $f\left(\mathrm{-2}\right).$

To evaluate $f\left(\mathrm{-2}\right),$ substitute $x=\mathrm{-2}.$
Simplify.
	The remainder is 5 when $f(x)=x^3+3x+19$ is divided by $x+2.$
Check: Use synthetic division to check.
The remainder is 5.

When we divided $8a^3+27$

by $2a+3$

in [link] the result was $4a^2-6a+9.$

To check our work, we multiply $4a^2-6a+9$

by $2a+3$

to get $8a^3+27$

.

Written this way, we can see that $4a^2-6a+9$

and $2a+3$

are factors of $8a^3+27.$

When we did the division, the remainder was zero.

Whenever a divisor, $x-c,$

divides a polynomial function, $f\left(x\right),$

and resulting in a remainder of zero, we say $x-c$

is a factor of $f\left(x\right).$

The reverse is also true. If $x-c$

is a factor of $f\left(x\right)$

then $x-c$

will divide the polynomial function resulting in a remainder of zero.

We will state this in the Factor Theorem.

Key Concepts

• Division of a Polynomial by a Monomial
  • To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
• Division of Polynomial Functions
  • For functions $f\left(x\right)$

    and

    $g(x),$

    where

    $g(x)\ne 0,$

  ---

  $\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$
• Remainder Theorem
  • If the polynomial function $f\left(x\right)$

    is divided by

    $x-c,$

    then the remainder is

    $f\left(c\right).$
• Factor Theorem: For any polynomial function $f\left(x\right),$
  • if $x-c$

    is a factor of

    $f\left(x\right),$

    then

    $f\left(c\right)=0$
  • if $f\left(c\right)=0,$

    then

    $x-c$

    is a factor of

    $f\left(x\right)$

Section Exercises

Practice Makes Perfect

Divide Monomials

In the following exercises, divide the monomials.

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Chapter Review Exercises

Add and Subtract Polynomials

Determine the Degree of Polynomials

In the following exercises, determine the type of polynomial.

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

In the following exercises, simplify.

Evaluate a Polynomial Function for a Given Value of the Variable

In the following exercises, find the function values for each polynomial function.

Add and Subtract Polynomial Functions

In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)

Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

In the following exercises, simplify each expression using the Product Property.

In the following exercises, simplify each expression using the Power Property.

In the following exercises, simplify each expression using the Product to a Power Property.

In the following exercises, simplify each expression using the Quotient to a Power Property.

In the following exercises, simplify each expression by applying several properties.

In the following exercises, write each number in scientific notation.

In the following exercises, convert each number to decimal form.

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

Multiply Polynomials

Multiply Monomials

In the following exercises, multiply the monomials.