6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 379 L F O I Factoring Trinomials of the Form ax bx c a , 1 2 + + ≠ Now we discuss how to factor an expression of the form ax bx c, 2 + + where a, the coefficient of the squared term, is not equal to 1. Consider the multiplication problem x x (2 1)( 3). + + + + = ⋅+ ⋅+⋅+⋅ = + + + = + + x x x x x x x x x x x (2 1)( 3) 2 2 3 1 1 3 2 6 3 2 7 3 2 2 Since x x x x (2 1)( 3) 2 7 3, 2 + + = + + the factors of x x 2 7 3 2 + + are x2 1 + and x 3. + Let’s study the coefficients more closely. F O I L x x 2 7 3 2 ↓ ↓ ↓ + + + x x (2 1)(1 3) + + = ⋅ = + = ⋅ + ⋅ = = ⋅ = F O I L 2 1 2 (2 3) (1 1) 7 1 3 3 Note that the product of the coefficients of the first terms in the multiplication of the binomials equals 2, the coefficient of the squared term. The sum of the products of the coefficients of the outer and inner terms equals 7, the coefficient of the x-term. The product of the last terms equals 3, the constant. A procedure to factor expressions of the form ax bx c a, 1, 2 + + ≠ follows. Learning Catalytics Keyword: Angel-SOM-6.9 (See Preface for additional details.) Example 4 Factoring a Trinomial Factor x x6 16. 2 − − Solution We must determine two numbers whose product is 16 − and whose sum is 6. − Begin by listing the factors of 16. − Factors of 16 − Sum of Factors 16(1) − 16 1 15 − + = − 8(2) − 8 2 6 − + = − 4(4) − 4 4 0 − + = 2(8) − 2 8 6 − + = 1(16) − 1 16 15 − + = The table lists all the factors of 16. − The only factors of 16 − whose sum is 6− are 8− and 2. We listed all factors so that you could see, for example, that 8(2) − is a different set of factors than 2(8). − Once you determine the factors you are looking for, there is no need to go any further. The trinomial can be written in factored form as − − = + − + − + x x x x x x 6 16 ( ( 8))( 2) or ( 8)( 2) 2 7 Now try Exercise 15
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