378 CHAPTER 6 Algebra, Graphs, and Functions expression as a product of its factors. For example, to factor x x 11 28 2 + + we write x x x x 11 28 ( 4)( 7) 2 + + = + + Let’s look at the factors more closely. x x x x 47 11 47 28 11 28 ( 4)( 7) 2 + = ⋅ = + + = + + Note that the sum of the two numbers in the factors is 4 7, + or 11. The 11 is the coefficient of the x-term. Also note that the product of the numbers in the two factors is 4 7, ⋅ or 28. The 28 is the constant in the trinomial. In general, when factoring an expression of the form x bx c, 2 + + we need to determine two numbers whose product is c and whose sum is b. When we determine the two numbers, the factors will be of the form + + ↑ ↑ x x ( ) ( ) . . One number Other number Example 3 Factoring a Trinomial Factor x x6 8. 2 + + Solution We need to determine two numbers whose product is 8 and whose sum is 6. Since the product is 8, + the two numbers must both be positive or both be negative. Because the coefficient of the x-term is positive, only the positive factors of 8 need to be considered. We begin by listing the positive numbers whose product is 8. Factors of 8 Sum of Factors 1(8) 1 8 9 + = 2(4) 2 4 6 + = Since 2 4 8 ⋅ = and 2 4 6, + = the numbers we are seeking are 2 and 4. Thus, we write x x x x 6 8 ( 2)( 4) 2 + + = + + Note that x x ( 4)( 2) + + is also an acceptable answer. When we are factoring a trinomial, we can always check our answer by multiplying the factors using the FOIL method. Do this now to check our factoring. 7 Now try Exercise 13 FACTORING TRINOMIALS OF THE FORM + + x bx c 2 1. Determine two numbers whose product is c and whose sum is b. 2. Write factors in the form + + ↑ ↑ x x ( ) ( ) . . 3. Check your answer by multiplying the factors using the FOIL method. PROCEDURE One number from Step 1 Other number from Step 1
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