More About Factoring Trinomials
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1 Section 6.3 More About Factoring Trinomials x 2 17x 70 x 7 x 10 Width of rectangle: Length of rectangle: x 7 x 10 Width of shaded region: 7 Length of shaded region: x 10 x 10 Area of shaded region: square units 85. The constant 3 in the trinomial x 2 4x 3 is positive, so the signs of the constants in the binomial factors must be the same. It is unnecessary to test x 1 x 3 or x 1 x 3 because both of these factorizations would yield a negative constant term in the product. The correct factorization is x 1 x A prime trinomial is a polynomial with three terms that cannot be factored with integer coefficients. 89. The process of factoring x 2 bx c is easier if c is a prime number because there are fewer factorizations to examine. Section 6.3 More About Factoring Trinomials 1. 5x 2 18x 9 x 3 5x a 2 12a 9 a 3 5a y 2 3y 27 y 3 2y 9 The missing factor is 5x 3. The missing factor is 5a 3. The missing factor is 2y z 2 13z 3 z 3 4z 1 9. The missing factor is 4z 1. 5x 3 x 1 5x 3 x 1 5x 1 x 3 5x 1 x x 12 x 1 5x 12 x x 2 5x 3 2x 3 x 1 5x 1 x 12 5x 1 x 12 5x 6 x 2 5x 6 x 2 5x 2 x 6 5x 2 x 6 5x 4 x 3 5x 4 x 3 5x 3 x 4 5x 3 x y 2 5y 1 4y 1 y y 2 3y 1 2y 1 y x 2 x 3 2x 3 x x 2 2x x 2 x s 2 10s x 2 13x 12 4x 3 x x 2 18x 8 3x 2 3x u 2 9u 2 6u 1 3u a 2 14a 8 5a 2 3a t 2 3t 18 5t 6 2t m 2 16m 15 5m 3 3m z 2 34z 15 8z 5 2z x 2 x 3 1 2x 2 x 3 2x 3 x 1 or 2x 3 x 1 or 2x 3 x 1
2 240 Chapter 6 Factoring and Solving Equations x 3x 2 2 3x 2 x or x 3x x 3x 2 1 3x 2 4x 4 3x 2 x 2 or 3x 2 x 2 or 3x 2 x 2 6x 2 7x x 2 7x 10 6x 5 x 2 or 6x 5 x 2 or 6x 5 x x 60x x 1 6x or x 60x x 2 4x 1 10x 1 6x 1 or 10x 1 6x 1 or 10x 1 6x x 15x 2 4 3x 4 5x or 16 8x 15x 2 15x 2 8x x 2 8x 16 5x 4 3x 4 or 5x 4 3x 4 or 5x 4 3x x 2 3x 3x 2x y 2 18y 3y 5y u u 3 9 u 3 u 3 u v 2 8v 42 2 v 2 4v 21 2 v 7 v x 2 3x 60 3 x 2 x z 2 24z z 2 8z 5 3 3z 5 z x 2 4x 2 2 2x 2 2x x 4 2x 3 8x 2 x 2 15x 2 2x 8 x 2 3x 2 5x x 3 4x 2 2x x 3x 2 4x x 3 24x 2 192x 6x x 2 4x 32 6x x 8 x u 4 18u 3 27u 2 9u 2 2u 2 2u b 13 b 13 b 31 b 31 b 17 b 17 3x 10 x 1 3x 10 x 1 3x 1 x 10 3x 1 x 10 3x 5 x 2 3x 5 x 2 3x 2 x 5 3x 2 x b 4 b 4 b 1 b 1 2 x 3 x 1 or 2x 6 x 1 or 2x 2 x 3 2 x 3 x 1 or 2x 6 x 1 or 2x 2 x 3 2x 1 x 6 2x 1 x 6 2x 3 x 2 2x 3 x 2
3 Section 6.3 More About Factoring Trinomials b 26 6x 20 x 1 or 3x 10 2x b 26 b 22 b 22 b 34 b 34 b 62 b 62 b 121 b 121 b 29 b 29 b 43 b 43 b 23 b 23 6x 20 x 1 or 3x 10 2x 2 6x 10 x 2 or 3x 5 2x 4 6x 10 x 2 or 3x 5 2x 4 6x 4 x 5 or 3x 2 2x 10 6x 4 x 5 or 3x 2 2x 10 6x 2 x 10 or 3x 1 2x 20 6x 2 x 10 or 3x 1 2x 20 6x 1 x 20 6x 1 x 20 6x 5 x 4 6x 5 x 4 3x 20 2x 1 3x 20 2x 1 3x 4 2x 5 3x 4 2x 5 c 1 c 10 c 27 c 7 c 22 c 45 4x 1 x 1 4x 5 x 2 4x 9 x 3 4x 7 x 1 4x 11 x 2 4x 15 x c 8 3x 4 x c 7 c 3 c 8 c 25 c 13 3x 7 x 1 3x 1 x 3 3x 2 x 4 3x 5 x 5 3x 13 x 1 c 1 c 14 c 11 c 1 c 6 c 4 6x 1 x 1 6x 7 x 2 6x 11 x 1 3x 1 2x 1 3x 2 2x 3 3x 4 2x ac b 7 The two numbers with a product of 6 and a sum of 7 are 6 and 1. 3x 2 7x 2 3x 2 6x x 2 3x 2 6x x 2 3x x 2 x 2 x 2 3x 1 ac b 1 The two numbers with a product of 6 and a sum of 1 are 3 and 2. 2x 2 x 3 2x 2 3x 2x 3 2x 2 3x 2x 3 x 2x 3 2x 3 2x 3 x ac ac b 5 The two numbers with a product of 24 and a sum of 5 are 3 and 8. 6x 2 5x 4 6x 2 3x 8x 4 The two numbers with a product of 30 and a sum of 11 are 6 and 5. 15x 2 11x 2 15x 2 6x 5x 2 6x 2 3x 8x 4 15x 2 6x 5x 2 3x 2x 1 4 2x 1 3x 5x 2 1 5x 2 2x 1 3x 4 5x 2 3x 1
4 242 Chapter 6 Factoring and Solving Equations 93. ac The two numbers with a product of 30 and a sum of 11 are 6 and 5. 3a 2 11a 10 3a 2 6a 5a 10 3a a 2 5 a 2 a 2 3a 5 ac b 2 The two numbers with a product of 48 and a sum of 2 are 8 and 6. 16x 2 2x 3 16x 2 8x 6x 3 16x 2 8x 6x 3 8x 2x 1 3 2x 1 2x 1 8x ac b 17 The two numbers with a product of 72 and a sum of 17 are 9 and 8. 12x 2 17x 6 12x 2 9x 8x 6 12x 2 9x 8x 6 3x 4x 3 2 4x 3 4x 3 3x 2 ac b 5 The two numbers with a product of 84 and a sum of 5 are 12 and 7. 6u 2 5u 14 6u 2 12u 7u 14 6u 2 12u 7u 14 6u u 2 7 u 2 u 2 6u x 2 5x 2 2x 1 x Volume Length Width Height 2x + 1 2x 3 7x 2 6x x 2x 2 7x 6 x x 1 x 2x 3 x 2 x 2x 3 x 2 x x Thus, 2x 3 is the length of the box x 2 9x 10 2x 5 x (a) y 1 2x 3 3x 2 5x Length of largest rectangle: 2x 5 x 2x 2 3x 5 Width of largest rectangle: x 2 x 2x 5 x 1 Length of shaded region: 2x 5 x x 5 y 1 y 2 Width of shaded region: 2 (b) 9 Area of shaded region: x 5 2 2x (c) The x-intercepts are 0, 0, 1, 0, and 5 2, 0. The y-intercept is 0, 0.
5 Mid-Chapter Quiz for Chapter (a) (b) Volume 12x 3 64x 2 48x 4x 3x 2 16x 12 Height Area of Base The area of the base of the bin is 3x 2 16x 12. Volume 12x 3 64x 2 48x 4x 3x 2 16x 12 4x 3x 2 x 6 The dimensions of the base of the bin must be 3x 2 and x The constant of the trinomial is 15, but the product of the last terms of the binomials is Four These are the four factorizations that need to be tested: ax 1 x c ax c x 1 ax 1 x c ax c x Examples of third-degree trinomials that have a common factor of 2x: 2x x 2 x 1 2x 3 2x 2 2x 2x x 2 7x 3 2x 3 14x 2 6x 2x x 5 x 1 2x x 2 4x 5 2x 3 8x 2 10x 2x x 3 x 3 2x x 2 6x 9 2x 3 12x 2 18x 2x 3x 7 x 2 2x 3x 2 13x 14 6x 3 26x 2 28x Mid-Chapter Quiz for Chapter x x x 3 2. x 2 y xy 2 xy x y 3. The missing factor is x y. y 2 y 42 y 7 y 6 The missing factor is y 6. The missing factor is 2x x 2 x 1 x 1 2x x x a 3 b 4a 2 b 2 2a 2 b a 2b The missing factor is 2x x x 2 3 x 2 x 2 x 3 8. t 3 3t 2 t 3 t 2 t 3 t 3 t 2 t 3 1 t 3 t 3 t y 2 11y 30 y 6 y u 2 u 30 u 6 u x 3 x 2 30x x x 2 x 30 x x 6 x x 2 y 8xy 64y 2y x 2 4x v 2 4v 2 2y x 8 x 4
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