380 CHAPTER 6 Algebra, Graphs, and Functions Remember, our factoring of trinomials may be checked by using the FOIL method of multiplication. We will check the results to Example 5: + + = ⋅+ ⋅+⋅+⋅ = + + + = + + x x x x x x x x x x x (3 2)( 5) 3 3 5 2 2 5 3 15 2 10 3 17 10 2 2 Since we obtained the expression we started with, our factoring is correct. FACTORING TRINOMIALS OF THE FORM + + ≠ ax bx c a , 1 2 1. Write all pairs of factors of the coefficient of the squared term, a. 2. Write all pairs of factors of the constant, c. 3. Try various combinations of these factors until the sum of the products of the outer and inner terms is bx. 4. Check your answer by multiplying the factors using the FOIL method. PROCEDURE Example 5 Factoring a Trinomial, 1 a ≠ Factor x x 3 17 10. 2 + + Solution The only positive factors of 3 are 3 and 1. Therefore, we write + + = x x x x 3 1710(3 )( ) 2 The number 10 has both positive and negative factors. However, since both the constant, 10, and the sum of the products of the outer and inner terms, 17, are positive, the two factors must be positive. The positive factors of 10 are 1(10) and 2(5). The following is a list of the possible factors. Possible Factors Sum of Products of Outer and Inner Terms x x (3 1)( 10) + + x 31 x x (3 10)( 1) + + x 13 x x (3 2)( 5) + + ←x 17 Correct middle term x x (3 5)( 2) + + x 11 Thus, x x x x 3 17 10 (3 2)( 5). 2 + + = + + 7 Now try Exercise 25 Example 6 Factoring a Trinomial, a 1 ≠ Factor x x 6 11 10. 2 − − Solution The factors of 6 will be either 6 1⋅ or 2 3. ⋅ Therefore, the factors may be of the form x x (6 )( ) or x x (2 )(3 ). When there is more than one set of factors for the first term, we generally try the medium-sized factors first. If that does not work, we try the other factors. Thus, we write x x x x 6 11 10(2 )(3 ) 2 − − = The factors of 10 − are ( 1)(10), (1)( 10), ( 2)(5), − − − and (2)( 5). − The correct factoring is x x x x 6 11 10 (2 5)(3 2). 2 − − = − + 7 Now try Exercise 33 Timely Tip To factor a trinomial, + + ax bx c, 2 first observe the sign of the constant term, c. x x x x x x x x c x x x x x x x x c 11 28 ( 7)( 4) 11 28 ( 7)( 4) If is positive, numbers have the same sign 3 28 ( 7)( 4) 3 28 ( 7)( 4) If is negative, numbers have different signs 2 2 2 2 + + → + + ↑ ↑ ↑ − + → − − ↑ ↑ ↑ + − → + − ↑ ↑ ↑ − − → − + ↑ ↑ ↑
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