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Factoring Perfect Square Trinomials

6.6.2 Factoring Perfect Square Trinomials

6.6.2 • Factoring Perfect Square Trinomials

Activity

Remember the patterns for perfect square trinomials:

a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 + 2 a b + b 2 = ( a + b ) 2

a 2 − 2 a b + b 2 = ( a − b ) 2 a 2 − 2 a b + b 2 = ( a − b ) 2

Factor each perfect square trinomial. If necessary, factor out the GCF first.

1. 64 y 2 − 80 y + 25 64 y 2 − 80 y + 25

Compare your answer:

( 8 y − 5 ) 2 ( 8 y − 5 ) 2

2. 16 y 2 + 24 y + 9 16 y 2 + 24 y + 9

Compare your answer:

( 4 y + 3 ) 2 ( 4 y + 3 ) 2

3. 25 n 2 − 120 n + 144 25 n 2 − 120 n + 144

Compare your answer:

( 5 n − 12 ) 2 ( 5 n − 12 ) 2

4. 100 y 2 − 20 y + 1 100 y 2 − 20 y + 1

Compare your answer:

( 10 y − 1 ) 2 ( 10 y − 1 ) 2

5. 10 j k 2 + 80 j k + 160 j 10 j k 2 + 80 j k + 160 j

Compare your answer:

10 j ( k + 4 ) 2 10 j ( k + 4 ) 2

6. 75 u 4 − 30 u 3 v + 3 u 2 v 2 75 u 4 − 30 u 3 v + 3 u 2 v 2

Compare your answer:

3 u 2 ( 5 u − v ) 2 3 u 2 ( 5 u − v ) 2

Are you ready for more?

Extending Your Thinking

Factor this perfect square trinomial.

90 p 4 r 2 + 300 p 3 q r 2 + 250 p 2 q 2 r 2 90 p 4 r 2 + 300 p 3 q r 2 + 250 p 2 q 2 r 2

Compare your answer:

10 p 2 r 2 ( 3 p + 5 q ) 2 10 p 2 r 2 ( 3 p + 5 q ) 2

Self Check

Factor: 64 m 2 + 112 m n + 49 n 2 .
  1. ( 8 m + 7 n ) 2
  2. ( 8 m − 7 n ) 2
  3. ( 8 n + 7 m ) 2
  4. m ( 8 + 7 n ) 2

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.

( a + b ) 2 ( a + b ) 2

( 3 x + 4 ) 2 ( 3 x + 4 ) 2

a 2 + 2 · a · b + b 2 ( 3 x ) 2 + 2 ( 3 x · 4 ) + 4 2 a 2 + 2 · a · b + b 2 ( 3 x ) 2 + 2 ( 3 x · 4 ) + 4 2

9 x 2 + 24 x + 16 9 x 2 + 24 x + 16

The trinomial 9 x 2 + 24 x + 16 9 x 2 + 24 x + 16 is called a perfect square trinomial. It is the square of the binomial 3 x + 4 3 x + 4 .

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 a x 2 + b x + c a x 2 + b x + c . 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 a a and b b are real numbers

a 2 + 2 a b + b 2 = ( a + b ) 2 a 2 + 2 a b + b 2 = ( a + b ) 2

a 2 − 2 a b + b 2 = ( a − b ) 2 a 2 − 2 a b + b 2 = ( a − b ) 2

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, a 2 a 2 . Next, check that the last term is a perfect square, b 2 b 2 . Then check the middle term: Is it the product, 2 a b 2 a b ? If everything checks, you can easily write the factors.

Example 1

Factor: 9 x 2 + 12 x + 4 9 x 2 + 12 x + 4 .

Step 1 - Does the trinomial fit the perfect square trinomials pattern, a 2 + 2 a b + b 2 a 2 + 2 a b + b 2 ?

  • Is the first term a perfect square? Write it as a square, a 2 a 2 .
    • Is 9 x 2 9 x 2 a perfect square?
    • Yes. Write it as ( 3 x ) 2 ( 3 x ) 2 .
  • Is the last term a perfect square? Write it as a square, b 2 b 2 .
    • Is 4 a perfect square?
    • Yes. Write it as ( 2 ) 2 ( 2 ) 2 .
  • Check the middle term. Is it 2 a b 2 a b ?
    • Is 12 x 12 x twice the product of 3 x 3 x and 2 2 2 2 ? Does it match?
    • Yes, so we have a perfect square trinomial!
Mathematical breakdown of 9x squared + 12x + 4: (3x) squared + 2(3x)(2) + (2) squared, showing how the middle term, 12x, comes from 2(3x)(2); arrows point to the 12x term.

Step 2 - Write the square of the binomial.

9 x 2 + 12 x + 4 9 x 2 + 12 x + 4

a 2 + 2 · a · b + b 2 a 2 + 2 · a · b + b 2

( 3 x ) 2 + 2 · 3 x · 2 + 2 2 ( 3 x ) 2 + 2 · 3 x · 2 + 2 2

( a + b ) 2 ( a + b ) 2

( 3 x + 2 ) 2 ( 3 x + 2 ) 2

Step 3 - Check by multiplying.

( 3 x + 2 ) 2 ( 3 x + 2 ) 2

( 3 x ) 2 + 2 · 3 x · 2 + 2 2 ( 3 x ) 2 + 2 · 3 x · 2 + 2 2

9 x 2 + 12 x + 4 ✓ 9 x 2 + 12 x + 4 ✓

The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern a 2 − 2 a b + b 2 a 2 − 2 a b + b 2 , which factors to ( a − b ) 2 ( a − b ) 2 .

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 2 a b 2 a b ?

Step 2 - Write the square of the binomial.

Step 3 - Check by multiplying.

A diagram showing the expansion of (a + b) squared and (a − b) squared, breaking them into a squared, b squared, and plus minus 2ab, with arrows indicating the terms combining to form each squared expression.

We’ll work one now where the middle term is negative.

Example 2

Factor: 81 y 2 − 72 y + 16 81 y 2 − 72 y + 16 .

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 ( a − b ) 2 ( a − b ) 2 .

Step 1 - Are the first and last terms perfect squares?

81 y 2 − 72 y + 16 81 y 2 − 72 y + 16

( 9 y ) 2 ( 4 ) 2 ( 9 y ) 2 ( 4 ) 2

Step 2 - Check the middle term.

Two arrows point from (9y squared) and (4) squared to 2(9y)(4) below them, and below that is 72y.

Does it match ( a − b ) 2 ( a − b ) 2 ?

a 2 − 2 a b + b 2 a 2 − 2 a b + b 2

( 9 y ) 2 − 2 · 9 y · 4 + 4 2 ( 9 y ) 2 − 2 · 9 y · 4 + 4 2

Yes.

Step 3 - Write as the square of a binomial.

( 9 y − 1 ) 2 ( 9 y − 1 ) 2

Step 4 - Check by multiplying.

( 9 y − 1 ) 2 ( 9 y − 1 ) 2

( 9 y ) 2 − 2 · 9 y · 4 + 4 2 ( 9 y ) 2 − 2 · 9 y · 4 + 4 2

81 y 2 − 72 y + 16 ✓ 81 y 2 − 72 y + 16 ✓

The next example will be a perfect square trinomial with two variables.

Example 3

Factor: 36 x 2 + 84 x y + 49 y 2 36 x 2 + 84 x y + 49 y 2 .

Step 1 - Test each term to verify the pattern.

a 2 + 2 ( a ) ( b ) + b 2 a 2 + 2 ( a ) ( b ) + b 2

6 x 2 + 2 ( 6 x ) ( 7 y ) + ( 7 y ) 2 6 x 2 + 2 ( 6 x ) ( 7 y ) + ( 7 y ) 2

Step 2 - Factor.

( 6 x + 7 y ) 2 ( 6 x + 7 y ) 2

Step 3 - Check by multiplying.

( 6 x + 7 y ) 2 ( 6 x + 7 y ) 2

( 6 x ) 2 + 2 · 6 x · 7 y + ( 7 y ) 2 ( 6 x ) 2 + 2 · 6 x · 7 y + ( 7 y ) 2

36 x 2 + 84 x y + 49 y 2 ✓ 36 x 2 + 84 x y + 49 y 2 ✓

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: 100 x 2 y − 80 x y + 16 y 100 x 2 y − 80 x y + 16 y .

Step 1 - Is there a GCF? Yes, 4 y 4 y , so factor it out.

4 y ( 25 x 2 − 20 x + 4 ) 4 y ( 25 x 2 − 20 x + 4 )

Step 2 - Is this a perfect square trinomial? Verify the pattern.

a 2 − 2 a b + b 2 a 2 − 2 a b + b 2

4 y [ ( 5 x ) 2 − 2 · 5 x · 2 + 2 2 ] 4 y [ ( 5 x ) 2 − 2 · 5 x · 2 + 2 2 ]

Step 3 - Write as the square of a binomial.

4 y ( 5 x − 2 ) 2 4 y ( 5 x − 2 ) 2

Remember: Keep the factor 4 y 4 y in the final product.

Step 4 - Check by multiplying.

4 y ( 5 x − 2 ) 2 4 y ( 5 x − 2 ) 2

4 y [ ( 5 x ) 2 − 2 · 5 x · 2 + 2 2 ] 4 y [ ( 5 x ) 2 − 2 · 5 x · 2 + 2 2 ]

4 y ( 25 x 2 − 20 x + 4 ) 4 y ( 25 x 2 − 20 x + 4 )

100 x 2 y − 80 x y + 16 y ✓ 100 x 2 y − 80 x y + 16 y ✓

Try it

Factoring Perfect Square Trinomials

Factor.

1. 4 x 2 + 12 x + 9 4 x 2 + 12 x + 9

2. 49 x 2 + 84 x y + 36 y 2 49 x 2 + 84 x y + 36 y 2

3. 8 x 2 y − 24 x y + 18 y 8 x 2 y − 24 x y + 18 y

Here is how to factor perfect square trinomials.

1. ( 2 x + 3 ) 2 ( 2 x + 3 ) 2

2. ( 7 x + 6 y ) 2 ( 7 x + 6 y ) 2

3. 2 y ( 2 x − 3 ) 2 2 y ( 2 x − 3 ) 2