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messersmith_power_introductory_algebra_1e_ch4_7_10

2 Factor a Trinomial of the Form x2 bx c The process of factoring is the opposite of multiplying. Let’s see how this will help us understand how to factor a trinomial of the form x2 bx c. Multiply (x 4)(x 5) using FOIL. (x 4)(x 5) x2 5x 4x 4 5 Multiply using FOIL. x2 (5 4)x 20 Use the distributive property, x2 9x 20 and multiply 4 5. (x 4)(x 5) x2 9x 20 The product of 4 and 5 is 20. c The sum of 4 and 5 is 9. So, if we were asked to factor x2 9x 20, we need to think of two integers whose product is 20 and whose sum is 9. Those numbers are 4 and 5. The factored form of x2 9x 20 is (x 4)(x 5). Procedure Factoring a Polynomial of the Form x2 bx c To factor a polynomial of the form x2 bx c, fi nd two integers m and n whose product is c and whose sum is b. Then, x2 bx c (x m)(x n). 1) If b and c are positive, then both m and n must be positive. 2) If c is positive and b is negative, then both m and n must be negative. 3) If c is negative, then one integer, m, must be positive and the other integer, n, must be negative. You can check the answer by multiplying the binomials. The result should be the original polynomial. Does this look similar to the procedure in Example 1? Compare the chart you made for Objective 1 to the procedure. Factor, if possible. a) x2 7x 12 b) p2 9p 14 c) w2 w 30 d) a2 3a 54 e) c2 6c 9 f ) y2 11y 35 Solution a) To factor x2 7x 12, we must fi nd two integers whose product is 12 and whose sum is 7. Both integers will be positive. Factors of 12 Sum of the Factors 1 12 1 12 13 2 6 2 6 8 3 4 3 4 7 EXAMPLE 2 In-Class Example 2 Factor, if possible. a) n2 9n 20 b) t2 14t 33 c) x2 x 72 d) k2 11k 12 e) y2 10y 25 f) d2 6d 28 Answer: a) (n 4)(n 5) b) (t 11)(t 3) c) (x 9)(x 8) d) (k 12)(k 1) e) (y 5)(y 5) or (y 5)2 f) prime The numbers are 3 and 4. Therefore, x2 7x 12 (x 3)(x 4). Check: (x 3)(x 4) x2 4 x 3x 12 x2 7x 12 ✓ 400 CHAPTER 7 Factoring Polynomials www.mhhe.com/messersmith


messersmith_power_introductory_algebra_1e_ch4_7_10
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