1. Factoring Perfect Square Trinomials and the Difference of Two Squares
4.5
Factoring Perfect Square Trinomials and the Difference of Two Squares

2. Perfect Square Trinomials
A trinomial that is the square of a binomial is called a perfect square trinomial.

3. Perfect Square Trinomials
In the last chapter we learned a shortcut for squaring a binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
We can now use these special products to help us factor perfect square trinomials, by reversing the equations.

4. Factoring Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2

5. Factoring Perfect Square Trinomials
Example
Decide whether 16x2 – 8xy + y2 is a perfect square trinomial.
Two terms, 16x2 and y2, are squares.
16x2 = (4x)2 and y2 = (y)2
Twice the product of 4x and y is the opposite of the other term of the trinomial.
2(4x)(y) = 8xy, the opposite of -8xy
Thus, 16x2 – 8xy + y2 is a perfect trinomial square.
16x2 – 8xy + y = (4x – y)2

6. Factoring Perfect Square Trinomials
Example
Factor: 16x2 – 8xy + y
Since we have determined that this trimonial is a perfect square, we can factor it using a special product formula.
16x2 – 8xy + y = (4x)2 – 2 • 4x • y + (y)2
a2 – 2 • a •b + b2
= (4x – y)2
(a – b)2

7. Difference of Two Squares
A binomial is the difference of two squares if
both terms are squares and
the signs of the terms are different.
For example,
9x2 – 25y2
–c4 + d4

8. Difference of Two Squares
Factoring the Difference of Two Squares a2 – b2 = (a + b)(a – b)

9. Difference of Two Squares
Example
Factor: x2 – 9.
The first term is a square and the last term, 9, is a square and can be written as 32. The signs of each term are different, so we have the difference of two squares.
x2 – 9 = (x – 3)(x + 3).