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

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

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.

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

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

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

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

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

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).