# Polynomials

## Introduction to Factoring Polynomials


### Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.
In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
First we will find the greatest common factor of two numbers.
In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.
In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

### Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example,  as  in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:
Here, we will start with a product, like  and end with its factors,  To do this we apply the Distributive Property “in reverse”.
The form on the left is used to multiply. The form on the right is used to factor.
So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!
Notice that in , we used the word factor as both a noun and a verb:
The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
In the next example, we factor a variable from a binomial.
When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!
When the leading coefficient, the coefficient of the first term, is negative, we factor the negative out as part of the GCF.
Pay close attention to the signs of the terms in the next example.

### Key Concepts

1. Find the greatest common factor.

2. Distributive Property

3. Factor the greatest common factor from a polynomial.


### Section Exercises


### Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions 
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial 
In the following exercises, factor the greatest common factor from each polynomial.

### Everyday Math


### Writing Exercises


### Self Check

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



ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?


### Chapter Review Exercises


### Add and Subtract Polynomials

Identify Polynomials, Monomials, Binomials and Trinomials 
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials 
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials 
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials 
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value of the Variable 
In the following exercises, evaluate each polynomial for the given value.

### Use Multiplication Properties of Exponents

Simplify Expressions with Exponents 
In the following exercises, simplify.
Simplify Expressions Using the Product Property of Exponents 
In the following exercises, simplify each expression.
Simplify Expressions Using the Power Property of Exponents 
In the following exercises, simplify each expression.
Simplify Expressions Using the Product to a Power Property 
In the following exercises, simplify each expression.
Simplify Expressions by Applying Several Properties 
In the following exercises, simplify each expression.
Multiply Monomials 
In the following exercises, multiply the monomials.

### Multiply Polynomials

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 various methods.
Multiply a Trinomial by a Binomial 
In the following exercises, multiply using any method.

### Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents 
In the following exercises, simplify.
Simplify Expressions with Zero Exponents 
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property 
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties 
In the following exercises, simplify.
Divide Monomials 
In the following exercises, divide the monomials.

### Integer Exponents and Scientific Notation

Use the Definition of a Negative Exponent 
In the following exercises, simplify.
Simplify Expressions with Integer Exponents 
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation 
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form 
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation 
In the following exercises, multiply and write your answer in decimal form.

### Introduction to Factoring Polynomials

Find the Greatest Common Factor of Two or More Expressions 
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial 
In the following exercises, factor the greatest common factor from each polynomial.

### Chapter Practice Test

In the following exercises, simplify each expression.
In the following exercises, factor the greatest common factor from each polynomial.
In the following exercises, simplify, and write your answer in decimal form.