Factoring Worksheet With Examples Page 38
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50Factoring Perfect Square Trinomials
2
2
Words
If a trinomial can be written in the form a
+ 2ab + b
or
2
2
2
a
- 2ab + b
, then it can be factored as (a + b )
or as
2
(a - b )
, respectively.
2
2
2
2
2
2
a
+ 2ab + b
= (a + b )
and a
- 2ab + b
= (a - b )
Symbols
EXAMPLE
Factor Perfect Square Trinomials
Determine whether each trinomial is a perfect square trinomial. If so,
factor it.
2
a. 16 x
+ 32x + 64
2
2
Is the first term a perfect square?
Yes, 16 x
= (4x )
.
2
Is the last term a perfect square?
Yes, 64 = 8
.
Is the middle term equal to 2(4x)(8)?
No, 32x ≠ 2(4x)(8).
2
16 x
+ 32x + 64 is not a perfect square trinomial.
2
b. 9 y
- 12y + 4
2
2
= (3y )
Is the first term a perfect square?
Yes, 9 y
.
2
Yes, 4 = 2
Is the last term a perfect square?
.
Yes, 12y = 2(3y)(2).
Is the middle term equal to 2(3y)(2)?
2
- 12y + 4 is a perfect square trinomial.
9 y
2
2
2
2
2
9 y
- 12y + 4 = (3y )
- 2(3y)(2) + 2
Write as a
- 2ab + b
.
2
= (3y - 2 )
Factor using the pattern.
2
2
1
A.
- 24n + 144
1
B.
+ 9x + 81
n
x
You have learned various techniques for factoring polynomials. The Concept
Summary can help you decide when to use a specific technique.
Factoring Polynomials
Number
Factoring Technique
Example
of Terms
2
2
2
2 or more
greatest common factor
3x
+ 6x
- 15x = 3x( x
+ 2x - 5)
difference of
2
2
2
2
a
- b
= (a + b)(a - b)
4x
- 25 = (2x + 5)(2x - 5)
squares
2
2
2
2
2
perfect square
a
+ 2ab + b
= (a + b)
x
+ 6x + 9 = (x + 3)
2
2
2
2
2
trinomial
a
- 2ab + b
= (a - b)
4x
- 4x + 1 = (2x - 1 )
2
x
+ bx + c = (x + m)(x + n)
2
2
x
+ bx + c
x
- 9x + 20 = (x - 5)(x - 4)
3
when m + n = b and mn = c.
2
2
2
2
ax
+ bx + c = ax
+ mx + nx + c
6x
- x - 2 = 6x
+ 3x - 4x - 2
2
a x
+ bx + c
when m + n = b and mn = ac.
= 3x(2x + 1) - 2(2x + 1)
Then use factoring by grouping.
= (2x + 1)(3x - 2)
ax + bx + ay + by
3xy - 6y + 5x - 10
factoring by
= x(a + b) + y(a + b)
= (3xy - 6y) + (5x - 10)
4 or more
grouping
= (a + b)(x + y)
= 3y(x -2) + 5(x - 2)
= (x - 2)(3y + 5)
455
Lesson 8-6 Perfect Squares and Factoring
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