54 CHAPTER R Review of Basic Concepts (b) 12x + 7212x - 72 = 4x2 - 14x + 14x - 49 FOIL method = 4x2 - 49 Combine like terms. (c) r213r + 2213r - 22 = r219r2 - 6r + 6r - 42 FOIL method = r219r2 - 42 Combine like terms. = 9r4 - 4r2 Distributive property S Now Try Exercises 29 and 31. EXAMPLE 4 Using the FOIL Method to MultiplyTwo Binomials Find each product. (a) 16m+ 1214m- 32 (b) 12x + 7212x - 72 (c) r213r + 2213r - 22 SOLUTION First Last F O I L (a) 16m+ 1214m- 32 = 6m14m2 + 6m1-32 + 114m2 + 11-32 Inner Outer = 24m2 - 18m+ 4m- 3 Multiply. = 24m2 - 14m- 3 Combine like terms. In Example 4(a), the product of two binomials is a trinomial, while in Examples 4(b) and (c), the product of two binomials is a binomial. The product of two binomials of the forms x + y and x - y is a special product form called a difference of squares. The squares of binomials, 1x + y22 and 1x - y22, are also special product forms called perfect square trinomials. Special Products Product of the Sum and Difference 1x +y2 1x −y2 =x2 −y2 of Two Terms (+)+* Difference of squares Square of a Binomial 1x +y22 =x2 +2xy +y2 1x −y22 =x2 −2xy +y2 (+1+)+1+* Perfect square trinomials EXAMPLE 5 Using the Special Products Find each product. (a) 13p + 11213p - 112 (b) 15m3 - 3215m3 + 32 (c) 19k - 11r3219k + 11r32 (d) 12m+ 522 (e) 13x - 7y422 SOLUTION (a) 13p + 11213p - 112 = 13p22 - 112 1x + y21x - y2 = x2 - y2 = 9p2 - 121 Power rule: 1ab2m = ambm 13p22 = 32p 2, not 3p 2
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