55 R.5 Polynomials (b) 15m3 - 3215m3 + 32 = 15m322 - 32 1x - y21x + y2 = x2 - y2 = 25m6 - 9 Power rules (c) 19k - 11r3219k + 11r32 = 19k22 - 111r322 = 81k2 - 121r6 (d) 12m+ 522 = 12m22 + 212m2152 + 52 1x + y22 = x2 + 2xy + y2 = 4m2 + 20m+ 25 Power rule; Multiply. (e) 13x - 7y422 = 13x22 - 213x217y42 + 17y422 1x - y22 = x2 - 2xy + y2 = 9x2 - 42xy4 + 49y8 Power rule; Multiply. S Now Try Exercises 43, 45, 47, and 49. Be careful applying the power rules. CAUTION The square of a binomial has three terms. 1x + y22 = x2 + 2xy + y2 1x - y22 = x2 - 2xy + y2 Remember to include the middle term. See Examples 5(d) and (e). EXAMPLE 6 Multiplying More Complicated Binomials Find each product. (a) 313p - 22 + 5q4313p - 22 - 5q4 (b) 1x + y23 (c) 12a + b24 SOLUTION (a) 313p - 22 + 5q4313p - 22 - 5q4 = 13p - 222 - 15q22 Product of the sum and difference of two terms = 9p2 - 12p + 4 - 25q2 Square both quantities. (b) 1x + y23 = 1x + y221x + y2 a3 = a2 # a = 1x2 + 2xy + y221x + y2 Square x + y. = x3 + x2y + 2x2y + 2xy2 + xy2 + y3 Multiply. = x3 + 3x2y + 3xy2 + y3 Combine like terms. (c) 12a + b24 = 12a + b2212a + b22 a4 = a2 # a2 = 14a2 + 4ab + b2214a2 + 4ab + b22 Square each 2a + b. = 16a4 + 16a3b + 4a2b2 + 16a3b + 16a2b2 Distributive property + 4ab3 + 4a2b2 + 4ab3 + b4 = 16a4 + 32a3b + 24a2b2 + 8ab3 + b4 Combine like terms. S Now Try Exercises 53, 57, and 59. This does not equal x 3 + y 3.
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