Activity
Remember the patterns for perfect square trinomials:
Factor each perfect square trinomial. If necessary, factor out the GCF first.
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Extending Your Thinking
Factor this perfect square trinomial.
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Self Check
Additional Resources
Factoring Perfect Square Trinomials
Some trinomials are perfect squares. They result from multiplying a binomial by itself. We squared a binomial using the Binomial Squares pattern in a previous lesson.
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.
PERFECT SQUARE TRINOMIALS PATTERN
If and are real numbers
To use 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.
Example 1
Factor: .
Step 1 - Does the trinomial fit the perfect square trinomials pattern, ?
- Is the first term a perfect square? Write it as a square, .
- Is a perfect square?
- Yes. Write it as .
- Is the last term a perfect square? Write it as a square, .
- Is 4 a perfect square?
- Yes. Write it as .
- Check the middle term. Is it ?
- Is twice the product of and ? Does it match?
- Yes, so we have a perfect square trinomial!
Step 2 - Write the square of the binomial.
Step 3 - Check by multiplying.
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.
How to factor perfect square trinomials:
Step 1 - Does the trinomial fit the perfect square trinomials pattern?
- Is the first term a perfect square? Write it as a square.
- Is the last term a perfect square? Write it as a square.
- Check the middle term. Is it ?
Step 2 - Write the square of the binomial.
Step 3 - Check by multiplying.
We’ll work one now where the middle term is negative.
Example 2
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 .
Step 1 - Are the first and last terms perfect squares?
Step 2 - Check the middle term.
Does it match ?
Yes.
Step 3 - Write as the square of a binomial.
Step 4 - Check by multiplying.
The next example will be a perfect square trinomial with two variables.
Example 3
Factor: .
Step 1 - Test each term to verify the pattern.
Step 2 - Factor.
Step 3 - Check by multiplying.
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.
Example 4
Factor: .
Step 1 - Is there a GCF? Yes, , so factor it out.
Step 2 - Is this a perfect square trinomial? Verify the pattern.
Step 3 - Write as the square of a binomial.
Remember: Keep the factor in the final product.
Step 4 - Check by multiplying.
Try it
Try It: Factoring Perfect Square Trinomials
Factor.
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Here is how to factor perfect square trinomials.
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