Polynomials and Polynomial Functions
Dividing Polynomials
Learning Objectives
By the end of this section, you will be able to:
- Dividing monomials
- Dividing a polynomial by a monomial
- Dividing polynomials using long division
- Dividing polynomials using synthetic division
- Dividing polynomial functions
- Use the remainder and factor theorems
Before you get started, take this readiness quiz.
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.
Find the quotient:
When we divide monomials with more than one variable, we write one fraction for each variable.
Find the quotient:
Find the quotient:
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.
Find the quotient:
Be very careful to simplify by dividing out a common factor, and to simplify the variables by subtracting their exponents.
Find the quotient:
Find the quotient:
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 simplifies to
Now we will do this in reverse to split a single fraction into separate fractions. For example, can be written
This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where then We will use this to divide polynomials by monomials.
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
Find the quotient:
Find the quotient:
Find the quotient:
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:
| Write it as a long division problem.
Be sure the dividend is in standard form. |
|
| Divide by It may help to ask yourself, “What do I need
to multiply by to get ?” |
|
| Put the answer, in the quotient over the term.
Multiply times Line up the like terms under the dividend. |
|
| Subtract from
You may find it easier to change the signs and then add. Then bring down the last term, 20. |
|
|
Divide by It may help to ask yourself, “What do I need to multiply by to get ?” Put the answer, , in the quotient over the constant term. |
|
| Multiply 4 times | |
| Subtract from | |
| Check:
Multiply the quotient by the divisor.