### 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

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

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

The answer is

3.

Factor out common monomials:

We recognize

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

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

We factor it and get

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

This expression is now completely factored.

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

1.

When we factor out the common binomial we get

2.

When we factor out the common binomial we get

### Example

Factor completely:

First, notice that each term has

Next, factor the trinomial in the parenthesis. Since

Rewrite the trinomial using

The final factored answer is:

### Review

Factor completely.

- \begin{align*}2x^2+16x+30\end{align*}
- \begin{align*}5x^2-70x+245\end{align*}
- \begin{align*}-x^3+17x^2-70x\end{align*}
- \begin{align*}2x^4-512\end{align*}
- \begin{align*}25x^4-20x^3+4x^2\end{align*}
- \begin{align*}12x^3+12x^2+3x\end{align*}
- \begin{align*}12c^2-75\end{align*}
- \begin{align*}6x^2-600\end{align*}
- \begin{align*}-5t^2-20t-20\end{align*}
- \begin{align*}6x^2+18x-24\end{align*}
- \begin{align*}-n^2+10n-21\end{align*}
- \begin{align*}2a^2-14a-16\end{align*}

### Review (Answers)

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