What if you had a polynomial like with multiple factors? How could you factor it completely? After completing this Concept, you'll be able to factor out common monomials and binomials from polynomials like this one.

### Watch This

CK-12 Foundation: 0912S Factoring Polynomials Completely

The WTAMU Virtual Math Lab has a detailed page on factoring polynomials here: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm. This page contains many videos showing example problems being solved.

### Guidance

We say that a polynomial is **factored completely** when we can’t factor it any more. Here are some suggestions that you should follow to make sure that you factor completely:

- Factor all common monomials first.
- Identify special products such as difference of squares or the square of a binomial. Factor according to their formulas.
- If there are no special products, factor using the methods we learned in the previous sections.
- Look at each factor and see if any of these can be factored further.

#### Example A

*Factor the following polynomials completely.*

a)

b)

c)

**Solution**

a) Factor out the common monomial. In this case 6 can be divided from each term:

There are no special products. We factor as a product of two binomials:

The two numbers that multiply to 6 and add to -5 are -2 and -3, so:

If we look at each factor we see that we can factor no more.

The answer is .

b) Factor out common monomials:

We recognize as a difference of squares. We factor it as .

If we look at each factor we see that we can factor no more.

The answer is .

c) Factor out common monomials:

We recognize as a perfect square and factor it as .

If we look at each factor we see that we can factor no more.

The answer is .

#### Example B

Factor the following polynomials completely:

a)

b)

**Solution**

a) Factor out the common monomial. In this case, factor out -2 rather than 2. (It’s always easier to factor out the negative number so that the highest degree term is positive.)

We recognize expression in parenthesis as a difference of squares. We factor and get:

If we look at each factor we see that the first parenthesis is a difference of squares. We factor and get:

If we look at each factor now we see that we can factor no more.

The answer is .

b) Factor out the common monomial:

We recognize as a perfect square and we factor it as .

We look at each term and recognize that the term in parentheses is a difference of squares.

We factor it and get , which we can rewrite as .

If we look at each factor now we see that we can factor no more.

The final answer is .

**Factor out a Common Binomial**

The first step in the factoring process is often factoring out the common monomials from a polynomial. But sometimes polynomials have common terms that are binomials. For example, consider the following expression:

Since the term appears in both terms of the polynomial, we can factor it out. We write that term in front of a set of parentheses containing the terms that are left over:

This expression is now completely factored.

Let’s look at some more examples.

#### Example C

*Factor out the common binomials.*

a)

b)

**Solution**

a) has a common binomial of .

When we factor out the common binomial we get .

b) has a common binomial of .

When we factor out the common binomial we get .

Watch this video for help with the Examples above.

CK-12 Foundation: Factoring Polynomials Completely

### Vocabulary

- We say that a polynomial is
**factored completely**when we factor as much as we can and we are unable to factor any more.

### Guided Practice

*Factor completely: .*

**Solution:**

First, notice that each term has as a factor. Start by factoring out :

Next, factor the trinomial in the parenthesis. Since find : . Find the factors of 12 that add up to -7. Since 12 is positive and -7 is negative, the two factors should be negative:

Rewrite the trinomial using , and then factor by grouping:

The final factored answer is:

### Practice

Factor completely.