66 CHAPTER R Review of Basic Concepts (c) 1a + 2b22 - 4c2 = 1a + 2b22 - 12c22 Write as a difference of squares. = 31a + 2b2 + 2c431a + 2b2 - 2c4 Factor. = 1a + 2b + 2c21a + 2b - 2c2 (d) x2 - 6x + 9 - y4 = 1x2 - 6x + 92 - y4 Group terms. = 1x - 322 - y4 Factor the trinomial. = 1x - 322 - 1y222 Write as a difference of squares. = 31x - 32 + y2431x - 32 - y24 Factor. = 1x - 3 + y221x - 3 - y22 (e) y2 - x2 + 6x - 9 = y2 - 1x2 - 6x + 92 Factor out the negative sign, and group the last three terms. = y2 - 1x - 322 Write as a difference of squares. = 3y - 1x - 3243y + 1x - 324 Factor. = 1y - x + 321y + x - 32 Distributive property. S Now Try Exercises 59, 61, 65, and 69. Check by multiplying. Be careful with signs.This is a perfect square trinomial. CAUTION When factoring as in Example 5(e), be careful with signs. Inserting an open parenthesis following the minus sign requires changing the signs of all of the following terms. EXAMPLE 6 Factoring Sums or Differences of Cubes Factor each polynomial. (a) x3 + 27 (b) m3 - 64n3 (c) 8q6 + 125p9 SOLUTION (a) x3 + 27 = x3 + 33 Write as a sum of cubes. = 1x + 321x2 - 3x + 322 Factor. = 1x + 321x2 - 3x + 92 Apply the exponent. (b) m3 - 64n3 = m3 - 14n23 Write as a difference of cubes. = 1m- 4n23m2 + m14n2 + 14n224 Factor. = 1m- 4n21m2 + 4mn + 16n22 Multiply; 14n22 = 42n2 (c) 8q6 + 125p9 = 12q223 + 15p323 Write as a sum of cubes. = 12q2 + 5p32312q222 - 2q215p32 + 15p3224 Factor. = 12q2 + 5p3214q4 - 10q2p3 + 25p62 Simplify. S Now Try Exercises 73, 75, and 77.
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