Factoring Perfect Square Trinomials
Let’s see what happens when we expand (a + b)2.
| |
= (a + b)2 |
| First, we use the definition of exponential notation
to write (a + b)2 as the product of two binomials.
|
= (a + b)(a + b)
|
| Next, we multiply the binomials.
Finally, we simplify and combine like terms. |
= a2 + ab + ba + b2
= a2 + 2ab + b2 |
The expression a2 + 2ab + b2 is a perfect square trinomial.
This means it is the result of squaring a binomial.
Note the structure of the three terms in the perfect square trinomial:
• The first term, a2, is a perfect square.
• The last term, b2, is a perfect square.
• The middle term, 2ab, is twice the product of a and b.
When we recognize this pattern, we can immediately factor the trinomial
as the square of a binomial.
A similar pattern holds when the middle term is subtracted rather than
added.
Pattern —
To Factor a Perfect Square Trinomial
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 - 2ab + b2 = (a - b)(a - b) = (a - b)2
Here is a useful procedure to follow when factoring by patterns.
Procedure —
To Factor by Patterns
Step 1 Decide if the given polynomial fits a pattern.
Step 2 Identify a and b. Then substitute in the pattern and simplify.
To check the factorization, multiply the factors.
Example 1
Factor: 4x2 + 4x + 1
Solution
Step 1 Decide if the given polynomial fits a pattern.
The first term, 4x2, is a perfect square, (2x)2.
The last term, 1, is a perfect square, (1)2.
The middle term, 4x, is twice the product of 2x and 1.
Therefore, 4x2 + 4x + 1 is a perfect square trinomial.
Step 2 Identify a and b. Then substitute in the pattern and simplify.
| In the factoring pattern
for a perfect square
trinomial, substitute 2x
for a and 1 for b. |
a2 + 2ab + b2
(2x)2 + 2(2x)(1) + (1)2 |
= ( a + b)( a + b) = (2x + 1)(2x + 1) |
The result is:
4x2 + 4x + 1 = (2x + 1)(2x + 1).
This can also be written: 4x2 + 4x + 1 = (2x + 1)2.
To check the factorization, we multiply.
| Is
Is
Is |
(2x + 1)(2x + 1) 4x2 + 2x + 2x
+ 1
4x2 + 4x + 1 |
= 4x2 + 4x + 1 ?
= 4x2 + 4x + 1 ?
= 4x2 + 4x + 1 ? Yes |
Example 2
Factor: 25y2 - 40y + 16
Solution
Step 1 Decide if the given polynomial fits a pattern.
The first term, 25y2, is a perfect square, (5y)2.
The last term, 16, is a perfect square, (4)2.
The middle term, -40y, is the opposite of twice the product of 5y and 4.
Therefore, 25y2 - 40y + 16 is a perfect square trinomial.
Step 2 Identify a and b. Then substitute in the pattern and simplify.
| In the factoring pattern
for a perfect square
trinomial, substitute 5y
for a and 4 for b. |
a2 - 2ab + b2
(5y)2 + 2(5y)(4) + (4)2 |
= ( a - b)( a - b) = (5y - 4)(5y - 4) |
The result is:
25y2 - 40y + 16 = (5y - 4)(5y - 4).
This can also be written: 25y2 - 40y + 16 = (5y - 4)2.
You can multiply to check the factorization. We leave the check to you.
