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messersmith_power_introductory_algebra_1e_ch4_7_10

EXAMPLE 1 In-Class Example 1 Factor completely. a) 5r2 32r 12 b) 6z2 13z 6 c) 16a2 6ab b2 Answer: a) (5r 2)(r 6) b) (3z 2)(2z 3) c) (8a b)(2a b) Read the explanations, follow the examples, take notes, and complete the You Trys. In the previous section, we learned that we could factor 2x2 10x 8 by fi rst taking out the GCF of 2 and then factoring the trinomial. 2x2 10x 8 2(x2 5x 4) 2(x 4)(x 1) In this section, we will learn how to factor a trinomial like 2x2 11x 15 where we cannot factor out the leading coeffi cient of 2. 1 Factor ax2 bx c (a 1) by Grouping Sum is 11. To factor 2x2 11x 15, fi rst fi nd the product of 2 and 15. Then, fi nd two integers Product: 2 15 30 whose product is 30 and whose sum is 11. The numbers are 6 and 5. Rewrite the middle term, 11x, as 6x 5x, then factor by grouping. 2x2 11x 15 2x2 6x 5x 15 2x(x 3) 5(x 3) Take out the common factor from each group. (x 3)(2x 5) Factor out (x 3). Check: (x 3)(2x 5) 2x2 5x 6x 15 2x2 11x 15 ✓ Factor completely. a) 8k2 14k 3 b) 6c2 17c 12 c) 7x2 34xy 5y2 Solution a) Since we cannot factor out a GCF (the GCF 1), we begin with a new method. Sum is 14. 8k2 14k 3 Think of two integers whose product is 24 and whose sum is 14. The integers are 2 and 12. Rewrite the middle term, 14k, as 2k 12k. Factor by grouping. Product: 8 3 24 8k2 14k 3 8k2 2k 12k 3 2k(4k 1) 3(4k 1) Take out the common factor from each group. (4k 1)(2k 3) Factor out (4k 1). Check by multiplying: (4k 1)(2k 3) 8k2 14k 3 ✓ b) Sum is 17. 6c2 17c 12 Product: 6 12 72 Think of two integers whose product is 72 and whose sum is 17. (Both numbers will be negative.) The integers are 9 and 8. Rewrite the middle term, 17c, as 9c 8c. Factor by grouping. 6c2 17c 12 6c2 9c 8c 12 3c(2c 3) 4(2c 3) Take out the common factor from each group. (2c 3)(3c 4) Factor out (2c 3). Check: (2c 3)(3c 4) 6c2 17c 12 ✓ Write the procedure, in your own words, for factoring these polynomials by grouping. 406 CHAPTER 7 Factoring Polynomials www.mhhe.com/messersmith


messersmith_power_introductory_algebra_1e_ch4_7_10
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