Factoring
Factor Special Products
Learning Objectives
By the end of this section, you will be able to:
- Factor perfect square trinomials
- Factor differences of squares
- Factor sums and differences of cubes
Before you get started, take this readiness quiz.
We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.
Factor Perfect Square Trinomials
Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter.
The trinomial is called a perfect square trinomial. It is the square of the binomial
In this chapter, you will start with a perfect square trinomial and factor it into its prime factors.
You could factor this trinomial using the methods described in the last section, since it is of the form But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.
Here is the pattern—the reverse of the binomial squares pattern.
If a and b are real numbers
To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, Next check that the last term is a perfect square, Then check the middle term—is it the product, If everything checks, you can easily write the factors.
Factor:
Factor:
Factor:
The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern which factors to
The steps are summarized here.
We’ll work one now where the middle term is negative.
Factor:
The first and last terms are squares. See if the middle term fits the pattern of a perfect square trinomial. The middle term is negative, so the binomial square would be
| Are the first and last terms perfect squares? | |
| Check the middle term. | |
| Does it match Yes. | |
| Write as the square of a binomial. | |
| Check by multiplying:
|
Factor:
Factor:
The next example will be a perfect square trinomial with two variables.
Factor:
| Test each term to verify the pattern. | |
| Factor. | |
| Check by multiplying. |
Factor:
Factor:
Remember the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.
Factor:
| Is there a GCF? Yes, so factor it out. | |
| Is this a perfect square trinomial? | |
| Verify the pattern. | |
| Factor. |
Remember: Keep the factor 4y in the final product.
Check:
Factor:
Factor:
Factor Differences of Squares
The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomi