Factoring Perfect-Square Trinomials and Differences of Squares 5.4 Recognizing Perfect-Square Trinomials Factoring Perfect-Square Trinomials Recognizing Differences of Squares Factoring Differences of Squares Factoring Completely

Recognizing Perfect-Square Trinomials A trinomial that is the square of a binomial is called a perfect-square trinomial. A2 + 2AB + B2 = (A + B)2; A2 – 2AB + B2 = (A – B)2

Determine whether each of the following is a perfect-square trinomial Determine whether each of the following is a perfect-square trinomial. a) x2 + 8x + 16 b) t2  9t  36 c) 25x2 + 4  20x Solution

Factoring a Perfect-Square Trinomial A2 + 2AB + B2 = (A + B)2; A2 – 2AB + B2 = (A – B)2

Factor: a) x2 + 8x + 16 b) 25x2  20x + 4 Solution

Sage and Scribe Factor: 16a2  24ab + 9b2 Factor: 12a3 108a2 + 243a

Recognizing Differences of Squares An expression, like 25x2  36, that can be written in the form A2  B2 is called a difference of squares.

Determine whether each of the following is a difference of squares Determine whether each of the following is a difference of squares. a) 16x2  25 b) 36  y5 c) x12 + 49 Solution

Factoring a Difference of Squares A2 – B2 = (A + B)(A – B).

Factor: a) x2  9 b) y2  16w2 Solution

Factor 5x4  125. Solution

Sage and Scribe 25  36a12 b) 98x2  8x8

Tips for Factoring 1. Always look first for a common factor! If there is one, factor it out. 2. Be alert for perfect-square trinomials and for binomials that are differences of squares. Once recognized, they can be factored without trial and error. 3. Always factor completely. 4. Check by multiplying.