6.6 Divide Polynomials

Learning Objectives

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

  • Divide a polynomial by a monomial
  • Divide a polynomial by a binomial

Divide 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.

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.

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.

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.

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.

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

Divide a Polynomial by a Binomial

To 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 write the long divisionThe long division of 875 by 25.
We divide the first two digits, 87, by 25.25 fits into 87 three times. 3 is written above the second digit of 875 in the long division bracket.
We multiply 3 times 25 and write the product under the 87.The product of 3 and 25 is 75, which is written below the first two digits of 875 in the long division bracket.
Now we subtract 75 from 87.87 minus 75 is 12, which is written under 75.
Then we bring down the third digit of the dividend, 5.The 5 in 875 is brought down next to the 12, making 125.
Repeat the process, dividing 25 into 125.25 fits into 125 five times. 5 is written to the right of the 3 on top of the long division bracket. 5 times 25 is 125. 125 minus 125 is zero. There is zero remainder, so 25 fits into 125 exactly five times. 875 divided by 25 equals 35.

We check division by multiplying the quotient by the divisor.

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

35⋅25
875✓

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.

When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.

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 Example 6.86, 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 Example 6.84, Example 6.85, and Example 6.86. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in Example 6.87 will be x4x 2+5x−2. It is missing an x 3 term. We will add in 0x3 as a placeholder.

In Example 6.88, we will divide by 2a−3. As we divide we will have to consider the constants as well as the variables.

Media

Access these online resources for additional instruction and practice with dividing polynomials:

  • Divide a Polynomial by a Monomial
  • Divide a Polynomial by a Monomial 2
  • Divide Polynomial by Binomial

Section 6.6 Exercises

Practice Makes Perfect

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

Divide a Polynomial by a Binomial

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

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “divide a polynomial by a monomial,” and “divide a polynomial by a binomial.” The rest of the cells are blank.

ⓑ After reviewing this checklist, what will you do to become confident for all goals?