Find two positive factors of 24 whose sum is 10. Make an organized list.
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1 9.5 Study Guide For use with pages GOAL Factor trinomials of the form x 2 1 bx 1 c. EXAMPLE 1 Factor when b and c are positive Factor x x Find two positive factors of 24 whose sum is 10. Make an organized list. 9.5 of 24 Sum of factors 24, , , correct sum 2, The factors 6 and 4 have a sum of 10, so they are the correct values of p and q. x x (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 x 1 24 Simplify. Factor when b is negative and c is positive Factor w w 1 9. Because b is negative and c is positive, p and q must be negative. of 9 Sum of factors 29, (21) 5210 correct sum 23, (23) 526 The factors 29 and 21 have a sum of 210, so they are the correct values of p and q. w w (x 2 9)(x 2 1) Exercises for Examples 1 and 2 1. x x y 2 1 6y z 2 2 7z
2 9.5 Study Guide continued For use with pages 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, , (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 (x 1 7)(x 2 1) Exercises for Example 3 4. x x y 2 1 2y z 2 2 5z 2 36 EXAMPLE 4 Solve a polynomial equation Solve the equation h 2 2 4h h 2 2 4h 5 21 Write original equation. h 2 2 4h Subtract 21 from each side. (h 1 3)(h 2 7) 5 0 Factor left side. h or h 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 x. 57
3 9.6 Study Guide For use with pages 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 n 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 n (n 2 1)(5n 2 7) correct EXAMPLE 2 Factor when b is negative and c is negative Factor 3m 2 2 5m 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 (m 1 2)(3m 2 11) Exercises for Examples 1 and a a b 2 2 8b c 2 1 5c 2 14 correct
4 9.6 Study Guide continued For use with pages 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 (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 (x 2 3)(2x 2 3) 9.6 Exercises for Example r 2 2 7r s 2 1 8s t 2 1 6t
5 9.7 Study Guide For use with pages 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 r 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 q 2 5 5( q 2 ) Factor out common factor. 5 5[4 2 2 (5q) 2 ] Write q 2 as a 2 2 b (2 2 5q)(2 1 5q) Difference of two squares Exercises for Example 1 1. m n y z
6 9.7 Study Guide continued For use with pages EXAMPLE 2 Factor perfect square trinomials a. x x x 2 1 2(x)(7) Write as a 2 1 2ab 1 b 2. 5 (x 1 7) 2 Perfect square trinomial b. 144y y (12y) 2 1 2(12y)(5) Write as a 2 1 2ab 1 b 2. 5 (12y 1 5) 2 Perfect square trinomial c. 150z z (25z z 1 1) Factor out common factor. 5 6[(5z) 2 2 2(5z)(1) ] Write 25z z 1 1 as a 2 1 2ab 1 b (5z 2 1) 2 Perfect square trinomial Exercises for Example 2 4. m } 2 m 1 1 } r rs 1 25s x x EXAMPLE 3 Solve a polynomial equation Solve the equation q q Write original equation. q Write left side as a 2 2 b 2. (q 1 10)(q 2 10) 5 0 Difference of two squares q or q 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 r m
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