Factoring Perfect Square Trinomials and the Difference of Two Squares

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

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

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).