65 R.6 Factoring Polynomials Make sure that the middle term of the trinomial being factored, -40pq here, is twice the product of the two terms in the binomial 4p - 5q. -40pq = 214p21-5q2 Thus, 16p2 - 40pq + 25q2 factors as 14p - 5q22. CHECK 14p - 5q22 = 16p2 - 40pq + 25q2 ✓ Multiply. (b) 36x2y2 + 84xy + 49 factors as 16xy + 7 2. CHECK Square 6xy + 7: 16xy + 722 = 36x2y2 + 84xy + 49. ✓ S Now Try Exercises 51 and 55. Factoring Binomials Check first to see whether the terms of a binomial have a common factor. If so, factor it out. The binomial may also fit one of the following patterns. 216xy2172 = 84xy Factoring Binomials Difference of Squares x2 −y2 = 1x +y2 1x −y2 Difference of Cubes x3 −y3 = 1x −y2 1x2 +xy +y22 Sum of Cubes x3 +y3 = 1x +y2 1x2 −xy +y22 CAUTION There is no factoring pattern for a sum of squares in the real number system. In particular, for real numbers x and y, x2 + y2 does not factor as 1x + y22. EXAMPLE 5 Factoring Differences of Squares Factor each polynomial. (a) 4m2 - 9 (b) 256k4 - 625m4 (c) 1a + 2b22 - 4c2 (d) x2 - 6x + 9 - y4 (e) y2 - x2 + 6x - 9 SOLUTION (a) 4m2 - 9 = 12m22 - 32 Write as a difference of squares. = 12m+ 3212m- 32 Factor. Check by multiplying. (b) 256k4 - 625m4 = 116k222 - 125m222 Write as a difference of squares. = 116k2 + 25m22116k2 - 25m22 Factor. = 116k2 + 25m2214k + 5m214k - 5m2 Factor 16k2 - 25m2. CHECK 116k2 + 25m2214k + 5m214k - 5m2 = 116k2 + 25m22116k2 - 25m22 Multiply the last two factors. = 256k4 - 625m4 ✓ Original polynomial Don’t stop here.
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