Find two positive factors of 24 whose sum is 10. Make an organized list.

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9.5 Study Guide For use with pages 582 589 GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x 2 1 10x 1 24. Find two positive factors of 24 whose sum is 10. Make an organized list. 9.5 of 24 Sum of factors 24, 1 24 1 1 5 25 12, 2 12 1 2 5 14 6, 4 6 1 4 5 10 correct sum 2, 3 2 1 3 5 5 The factors 6 and 4 have a sum of 10, so they are the correct values of p and q. x 2 1 10x 1 24 5 (x 1 6)(x 1 4) EXAMPLE 2 CHECK (x 1 6)(x 1 4) 5 x 2 1 4x 1 6x 1 24 Multiply binomials. 5 x 2 1 10x 1 24 Simplify. Factor when b is negative and c is positive Factor w 2 2 10w 1 9. Because b is negative and c is positive, p and q must be negative. of 9 Sum of factors 29, 21 29 1 (21) 5210 correct sum 23, 23 23 1 (23) 526 The factors 29 and 21 have a sum of 210, so they are the correct values of p and q. w 2 2 10w 1 9 5 (x 2 9)(x 2 1) Exercises for Examples 1 and 2 1. x 2 1 10x 1 16 2. y 2 1 6y 1 5 3. z 2 2 7z 1 12 56

9.5 Study Guide continued For use with pages 582 589 EXAMPLE 3 Factor when b is positive and c is negative Factor k 2 1 6x 2 7. Because c is negative, p and q must have different signs. of 7 Sum of factors 27, 1 27 1 1 526 7, 21 71 (21) 5 6 correct sum The factors 7 and 21 have a sum of 6, so they are the correct values of p and q. k 2 1 6k 2 7 5 (x 1 7)(x 2 1) Exercises for Example 3 4. x 2 2 10x 2 11 9.5 5. y 2 1 2y 2 63 6. z 2 2 5z 2 36 EXAMPLE 4 Solve a polynomial equation Solve the equation h 2 2 4h 5 21. h 2 2 4h 5 21 Write original equation. h 2 2 4h 2 21 5 0 Subtract 21 from each side. (h 1 3)(h 2 7) 5 0 Factor left side. h 1 3 5 0 or h 2 7 5 0 Zero-product property h523 or h 5 7 Solve for h. The roots of the equation are 23 and 7. Exercise for Example 4 7. Solve the equation x 2 1 30 5 11x. 57

9.6 Study Guide For use with pages 592 599 GOAL Factor trinomials of the form ax 2 1 bx 1 c. EXAMPLE 1 Factor when b is negative and c is positive Factor 5n 2 2 12n 1 7. Because b is negative and c is positive, both factors of c must be negative. Make a table to organize your work. You must consider the order of the factors of 7, because the x-terms of the possible factorization are different. of 5 of 7 Possible factorization Middle term when multiplied 1, 5 21, 27 (n 2 1)(5n 2 7) 25n 2 7n 5212n 1, 5 27, 21 (n 2 7)(5n 2 1) 2n 2 35n 5236n 5n 2 2 12n 1 7 5 (n 2 1)(5n 2 7) correct EXAMPLE 2 Factor when b is negative and c is negative Factor 3m 2 2 5m 2 22. Because b is negative and c is negative, p and q must have different signs. of 3 of 22 Possible factorization Middle term when multiplied 1, 3 1, 222 (m 1 1)(3m 2 22) 3m 2 22m 5219m 1, 3 21, 22 (m 2 1)(3m 1 22) 22m 2 3m 5 19m 1, 3 2, 211 (m 1 2)(3m 2 11) 211m 1 6m 525m 1, 3 211, 2 (m 2 11)(3m 1 2) 2m 2 33m 5231m 3m 2 2 5m 2 22 5 (m 1 2)(3m 2 11) Exercises for Examples 1 and 2 1. 7a 2 2 50a 1 7 2. 4b 2 2 8b 2 5 3. 6c 2 1 5c 2 14 correct 9.6 67

9.6 Study Guide continued For use with pages 592 599 EXAMPLE 3 Factor when a is negative Factor 22x 2 1 9x 2 9. STEP 1 Factor 21 from each term of the trinomial. 22x 2 1 9x 2 9 52(2x 2 2 9x 1 9) STEP 2 Factor the trinomial 2x 2 2 9x 1 9. Because b is negative and c is positive, both factors of c must be negative. Use a table to organize information about the factors of a and c. of 2 of 9 Possible factorization Middle term when multiplied 1, 2 21, 29 (x 2 1)(2x 2 9) 29x 2 2x 5211x 1, 2 29, 21 (x 2 9)(2x 2 1) 2x 2 18x 5219x 1, 2 23, 23 (x 2 3)(2x 2 3) 23x 2 6x 529x correct 22x 2 1 9x 2 9 52(x 2 3)(2x 2 3) 9.6 Exercises for Example 3 4. 23r 2 2 7r 2 4 5. 23s 2 1 8s 1 16 6. 28t 2 1 6t 2 1 68

9.7 Study Guide For use with pages 600 605 GOAL Factor special products. Vocabulary The for finding the square of a binomial gives you the for factoring trinomials of the form a 2 1 2ab 1 b 2 and a 2 2 2ab 1 b 2. These are called perfect square trinomials. EXAMPLE 1 Factor the difference of squares a. r 2 2 81 5 r 2 2 9 2 Write as a 2 2 b 2. 5 (r 2 9)(r 1 9) Difference of two squares b. 9s 2 2 4t 2 5 (3s) 2 2 (2t) 2 Write as a 2 2 b 2. 5 (3s 2 2t)(3s 1 2t) Difference of two squares c. 80 2 125q 2 5 5(16 2 25q 2 ) Factor out common factor. 5 5[4 2 2 (5q) 2 ] Write 16 2 25q 2 as a 2 2 b 2. 5 5(2 2 5q)(2 1 5q) Difference of two squares Exercises for Example 1 1. m 2 2 121 2. 9n 2 2 64 3. 3y 2 2 147z 2 9.7 77

9.7 Study Guide continued For use with pages 600 605 EXAMPLE 2 Factor perfect square trinomials a. x 2 1 14x 1 49 5 x 2 1 2(x)(7) 1 7 2 Write as a 2 1 2ab 1 b 2. 5 (x 1 7) 2 Perfect square trinomial b. 144y 2 2 120y 1 25 5 (12y) 2 1 2(12y)(5) 1 5 2 Write as a 2 1 2ab 1 b 2. 5 (12y 1 5) 2 Perfect square trinomial c. 150z 2 2 60z 1 6 5 6(25z 2 2 10z 1 1) Factor out common factor. 5 6[(5z) 2 2 2(5z)(1) 1 1 2 ] Write 25z 2 2 10z 1 1 as a 2 1 2ab 1 b 2. 5 6(5z 2 1) 2 Perfect square trinomial Exercises for Example 2 4. m 2 2 1 } 2 m 1 1 } 16 5. 16r 2 1 40rs 1 25s 2 6. 36x 2 2 36x 1 9 9.7 EXAMPLE 3 Solve a polynomial equation Solve the equation q 2 2 100 5 0. q 2 2 100 5 0 Write original equation. q 2 2 10 2 5 0 Write left side as a 2 2 b 2. (q 1 10)(q 2 10) 5 0 Difference of two squares q 1 10 5 0 or q 2 10 5 0 Zero-product property q 5210 or q 5 10 Solve for q. The roots of the equation are 210 and 10. Exercises for Example 3 Solve the equation. 7. r 2 2 10r 1 25 5 0 8. 16m 2 2 81 5 0 78

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