One of the factoring problems encountered most often in algebra is the factoring of trinomials. In this section, we will discuss how to factor trinomials like x2 11x 18, 3n2 n 4, 4a2 11ab 6b2, and many more. Let’s begin with trinomials of the form x2 bx c, where the coeffi cient of the squared term is 1. 1 Factor a Trinomial of the Form x2 bx c In Section 7.1 we said that 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 7) using FOIL. (x 4)(x 7) x2 7x 4x 4 7 Multiply using FOIL. x2 (7 4)x 28 Use the distributive property, and multiply 4 7. x2 11x 28 (x 4)(x 7) x2 11x 28 b R 11 is the sum of 4 and 7. 28 is the product of 4 and 7. So, if we were asked to factor x2 11x 28, we need to think of two integers whose product is 28 and whose sum is 11. Those numbers are 4 and 7. The factored form of x2 11x 28 is (x 4)(x 7). 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. EXAMPLE 1 Factor, if possible. a) y2 8y 15 b) k2 k 56 c) r2 7r 9 Solution a) To factor y2 8y 15, fi nd the two integers whose product is 15 and whose sum is 8. Since 15 is positive and the coeffi cient of y is a negative number, 8, both integers will be negative. Factors of 15 Sum of the Factors 1 (15) 15 1 (15) 16 3 (5) 15 3 (5) 8 The numbers are 3 and 5: y2 8y 15 ( y 3)( y 5). Check: ( y 3)( y 5) y2 5y 3y 15 y2 8y 15 ✓ We are finding two things. The first is the product of the factors, and the second is the sum of the factors. www.mhhe.com/messersmith SECTION 7.2 Factoring Trinomials 367
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