### Factoring Completely

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.

#### Factoring Completely - Learn by Example

1.

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 .

2.

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 .

3.

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 .

4.

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 .

5.

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.

#### Factoring out Common Binomials - Learn by Example

1.

has a common binomial of .

When we factor out the common binomial we get .

2.

has a common binomial of .

When we factor out the common binomial we get .

### Example

Factor completely: .

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:

### Review

Factor completely.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 9.12.