# Polynomials and Polynomial Functions

## 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  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.

### 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.
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  It is missing an  term. We will add in  as a placeholder.
In the next example, we will divide by  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  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  and  are removed. as well as the  and  as they are opposite the term above.
The first row of the synthetic division is the coefficients of the dividend. The  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.
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  the value of the function at  is the same as the remainder from the division problem.
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  we multiply the quotient,  times the divisor,  and add the remainder, r.


This leads us to the Remainder Theorem.
When we divided  by  in  the result was  To check our work, we multiply  by  to get .
Written this way, we can see that  and  are factors of  When we did the division, the remainder was zero.
Whenever a divisor,  divides a polynomial function,  and resulting in a remainder of zero, we say  is a factor of 
The reverse is also true. If  is a factor of  then  will divide the polynomial function resulting in a remainder of zero.
We will state this in the Factor Theorem.

### Key Concepts

1. Division of a Polynomial by a Monomial


2. Division of Polynomial Functions


3. Remainder Theorem


4. Factor Theorem: For any polynomial function 



### 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.
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
Divide Polynomial Functions
In the following exercises, divide.
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
In the following exercises, use the Factor Theorem to determine if  is a factor of the polynomial function.

### 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

Types 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.
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using:
ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Polynomial by a Polynomial
In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.
Multiply Special Products
In the following exercises, square each binomial using the Binomial Squares Pattern.
In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

### Divide Monomials

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.
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
Divide Polynomial Functions
In the following exercises, divide.
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
In the following exercises, use the Factor Theorem to determine if  is a factor of the polynomial function.

### Chapter Practice Test

In the following exercises, simplify and write your answer in exponential notation.