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# Factoring Completely

## Sum or difference with higher powers, factoring algorithm, and grouping.

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Factoring Completely

### 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.

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.

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.

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.

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.

#### 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:

### Review

Factor completely.

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Color Highlighted Text Notes

### Vocabulary Language: English

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = $2^3 = 2 \times 2 \times 2 = 8$.

factor

Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.

Factor to Solve

"Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of $x$ that make each binomial equal to zero.

The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

Trinomial

A trinomial is a mathematical expression with three terms.