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Formula Factoring the Difference of Two Squares a2 b2 (a b)(a b) Don’t forget that we can check all factorizations by multiplying. EXAMPLE 3 Factor completely. a) y2 9 b) 25m2 16n2 c) w2 9 64 d) c2 36 Solution a) First, notice that y2 9 is the difference of two terms and those terms are perfect squares. We can use the formula a2 b2 (a b)(a b). Identify a and b. y2 9 T T What do you square to get y2? y (y)2 (3)2 What do you square to get 9? 3 Then, a y and b 3. Therefore, y2 9 ( y 3)( y 3). b) Look carefully at 25m2 16n2. Each term is a perfect square, and they are being subtracted. Identify a and b. 25m2 16n2 T T What do you square to get 25m2? 5m (5m)2 (4n)2 What do you square to get 16n2? 4n Then, a 5m and b 4n. So, 25m2 16n2 (5m 4n)(5m 4n). c) Each term in w2 9 64 is a perfect square, and they are being subtracted. w2 9 64 T T What do you square to get w2? w (w)2 a3 8 b 2 What do you square to get 9 64 ? 3 8 So, a w and b 3 8 . Therefore, w2 9 64 aw 3 8 b aw 3 8 b. d) Each term in c2 36 is a perfect square, but the expression is the sum of two squares. This polynomial does not factor. c2 36 (c 6)(c 6) since (c 6)(c 6) c2 36. c2 36 (c 6)(c 6) since (c 6)(c 6) c2 12c 36. So, c2 36 is prime. Notice in this example that the coefficients of the first and last terms of the binomials are both perfect squares! www.mhhe.com/messersmith SECTION 7.3 Special Factoring Techniques 381


messersmith_power_intermediate_algebra_1e_ch4_7_10
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