62 CHAPTER R Review of Basic Concepts (b) 6x2t + 8xt - 12t = 2t13x2 + 4x - 62 GCF = 2t CHECK 2t13x2 + 4x - 62 = 6x2t + 8xt - 12t ✓ (c) 141m+ 1 3 - 281m+ 122 - 71m+ 12 = 71m+ 12321m+ 122 - 41m+ 12 - 14 GCF = 71m+ 12 = 71m+ 12321m2 + 2m+ 12 - 4m- 4 - 14 Square m+ 1; distributive property = 71m+ 1212m2 + 4m+ 2 - 4m- 4 - 12 Distributive property = 71m+ 1212m2 - 32 Combine like terms. S Now Try Exercises 13, 19, and 25. Remember the middle term. CAUTION In Example 1(a), the 1 is essential in the answer because y219y32 ≠9y5 + y2. Factoring can always be checked by multiplying. EXAMPLE 2 Factoring by Grouping Factor each polynomial by grouping. (a) mp2 + 7m+ 3p2 + 21 (b) 2y2 + az - 2z - ay2 (c) 4x3 + 2x2 - 2x - 1 SOLUTION (a) mp2 + 7m+ 3p2 + 21 = 1mp2 + 7m2 + 13p2 + 212 Group the terms. = m1p2 + 72 + 31p2 + 72 Factor each group. = 1p2 + 721m+ 32 p2 + 7 is a common factor. CHECK 1p2 + 721m+ 32 = mp2 + 3p2 + 7m+ 21 FOIL method = mp2 + 7m+ 3p2 + 21 ✓ Commutative property Factoring by Grouping When a polynomial has more than three terms, it can sometimes be factored using factoring by grouping. Consider this example. ax + ay + 6x + 6y Terms with common factor a Terms with common factor 6 (+)+* (+)+* = 1ax + ay2 + 16x + 6y2 Group the terms so that each group has a common factor. = a1x + y2 + 61x + y2 Factor each group. = 1x + y21a + 62 Factor out x + y. It is not always obvious which terms should be grouped. In cases like the one above, group in pairs. Experience and repeated trials are the most reliable tools.
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