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# Factoring Special Products

Perfect Square Trinomials

 Perfect Square Trinomials a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a − b)2

In order for a polynomial to be a perfect square
trinomial , two conditions must be satisfied:

1. The first and last terms must be perfect squares.

2. The “ middle term ” must equal 2 or – 2 times the
product of the expressions being squared in the
first and last term.

Example: Factor the following :

x2 + 8x + 16 = (x + 4)2

b.) 9x4 − 30x2z + 25z2

9x4 − 30x2z + 25z2 = (3x2 – 5z)2

The Difference of Two Squares

 Difference of Two Squares a2 – b2 = (a + b)(a − b)

Example: Factor the following :

a.) 3x2 − 27
3x2 − 27 = 3(x2 – 9)  3 is a common factor.
= 3(x + 3)(x – 3)

b.) 4x2 − 25y4
4x2 − 25y4 = (2x + 5y2)(2x – 5y2)

The Sum of Two Cubes

 The Sum of Two Cubes a3 + b3 = (a + b) (a2 − ab + b2)

Example: Factor: 27x3 + 125
a = 3x
b = 5

27x3 + 125 = (3x)3 + 53
= (3x + 5) [(3x)2 − 3x·5 + 52]
= (3x + 5)(9x2 − 15x + 25)

The Difference of Two Cubes

 The Difference of Two Cubes a3 − b3 = (a − b) (a2 + ab + b2)

Example: Factor: x3 − 64
a = x
b = 4

x3 − 64 = x3 – 43
= (x − 4) (x2 + 4x + 42)
= (x − 4) (x2 + 4x + 16)

Practice

Factor each perfect square trinomial completely

x2 + 14x + 49

x2 − 8x + 16

4a2 − 12ab + 9b2

4x2 + 20x + 25

n2 − 16

a4 − 16

36 p2 − 49q2

9x6 − 25y8

64x3 + 1

8x3 − 27

4x4 − 4xy3

125a6 + 8b3

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