Additional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping

Transcription

1 Additional Exercises 7.1 Form I The Greatest Common Factor and Factoring by Grouping Find the greatest common factor of each list of monomials x and 15 x y and 8y a 3 a, 4 and 4 3a x y, 15x y and 4 1x 4. Factor the greatest common factor from the polynomial. If there is no factor other then 1 and the polynomial cannot be factored, so state. 5. 5x a 3 a y + 5y + 6y x + 18x 1x x y 0x y + 1xy x y + 10x y 14x y 10. AE-7

2 Factor out the common binomial factor from each polynomial. 11. x ( y + 14) + 6( y + 14) a ( b 7) 3( b 7) ( x + 10) y( x + 10) 13. Factor by grouping. 14. xy + 4 y + x ab + 3b 6a xy 5y + x x 3 x + x ab + b + 9a xy + y 10x x x + x 0. AE-8

3 Additional Exercises 7.1 Form II The Greatest Common Factor and Factoring by Grouping Find the greatest common factor of each list of monomials. 1. 1x and 0 x y x and xy b 9 a and 88 b 5 9 a n m n, 18m and 30 n 3 m 4. Factor the greatest common factor from the polynomial. If there is no factor other then 1 and the polynomial cannot be factored, so state x 3 x x 5 7 x a a + a x y 44x y + 36x y x y 16x y 0x y x y 5x y + 55x y 60x y 10. AE-9

4 Factor out the common binomial factor from each polynomial. 11. x ( y 7) ( y 7) a ( b + 3) + 4( b + 3) a (8 + b) (8 + b) 13. Factor by grouping. 14. x 3 x 3x xy + 5 y + 4x b + ab + 4b + 8a xy + 4y + 15x xy x 1y x 8x 3x a b 16ab a b AE-30

5 Additional Exercises 7.1 Form III The Greatest Common Factor and Factoring by Grouping Find the greatest common factor of each list of monomials x and 4x x 8 6 x, 45 and 4 1x y x y, 30x and 84 y 5 5 x 3. Factor the greatest common factor from the polynomial. If there is no factor other then 1 and the polynomial cannot be factored, so state y 9y + 1y x xy x y a 15a + 5a x y + 40x y + 64x y m n 48n n 64mn 8. Factor out the common binomial factor from each polynomial. 9. x ( x 3) + 6( x 3) y ( x + 7) ( x + 7) 10. AE-31

6 11. 6x (x 7) + (x 7) 11. Factor by grouping x 3x 4x + 1x ab 4b + 6a 4b xy + 16y 5x mn + n 7m x 6x + 15x x 5x + 1x 15x a 16a 9a Solve. 19. The area of a rectangle is x + 6xy + 4xy + 1y. The width 19. of the rectangle is x + 3y. Write a polynomial for the length of the rectangle. 0. The width of a rectangle is x + y. The length of the rectangle 0. is x 5. Write a polynomial for the area of the rectangle. AE-3

7 Additional Exercises 7. Form I Factoring Trinomials Whose Leading Coefficient is 1 Factor each trinomial, or state that the trinomial is prime. 1. x + 5x x + 7x y + 8y a 6a x 8x y 8y a + a x 3x m 3m a + 4ab 1b 10. AE-33

8 11. a 6xy 7 y x 6xy 7y a 11a x + 11x Factor each trinomial completely. 15. x 4x a 15a x 3 + 7x + 6x x 3 y + x y 35xy x 3x a + 56a + 96a 0. AE-34

9 Additional Exercises 7. Form II Factoring Trinomials Whose Leading Coefficient is 1 Factor each trinomial, or state that the trinomial is prime. 1. x + 10x x 10x x x a + 10a y + 9y x 15x x + 5x a + 14a m + 6m x 11x AE-35

10 11. y + y a + 9a x 9x y 1y Factor each trinomial completely x + 10x x + 8x x x 48x a + 18a + 40a x 3 y + 3x y 40xy y + 66y + 148y 0. AE-36

11 Additional Exercises 7. Form III Factoring Trinomials Whose Leading Coefficient is 1 Factor each trinomial, or state that the trinomial is prime. 1. x + 14x a 5a y + 9y x 18x x 13x y 3y a + 15ab 16b x + 3x y 16y a 5a AE-37

12 11. x + + 1xy 54y y 8y a 11a x 11xy 60y 14. Factor each trinomial completely x 8x + 16x x 30x a + 7a + 60a x y + x y 4x y y 30y x + xy x y AE-38

13 Additional Exercises 7.3 Form I Factoring Trinomials Whose Leading Coefficient is Not 1 Use the method of your choice to factor each trinomial, or state that the trinomial is prime. 1. x + 9x x + 16x a + 11a x + x y y x 14x a 13a x 19x y 13y x + 6x AE-39

14 11. 10x 17x y + 4y a 34a Factor completely x + 6x x 66x a a a y + 8y + 4y x x 4 3 4x a 18a 18a x + 45x AE-40

15 Additional Exercises 7.3 Form II Factoring Trinomials Whose Leading Coefficient is Not 1 Use the method of your choice to factor each trinomial, or state that the trinomial is prime. 1. 6x + 11x x + 11x x + 7x x 14x x 5x x + x x + 33x x + 17x x 7x x 8x AE-41

16 11. 1x y + 5xy x + 34x x y 5xy Factor completely a + 1a a + 66a y 35y x + 18x + 4x a 1a 45a y + 68y 30y x 40x 3 0. AE-4

17 Additional Exercises 7.3 Form III Factoring Trinomials Whose Leading Coefficient is Not 1 Use the method of your choice to factor each polynomial, or state that the polynomial is prime. 1. 7x + 18x a a x + 7x x + 8x x + 14x a + 7a y + 9y x y + xy x x 6x b + 14b AE-43

18 11. 0x + 1x x y 14xy a 11a x + 35x a + 19a y 78y x + y + 35xy x 30x x y + 40x y + 5xy The area of a rectangle is 36x + 66x If the length is 0. 6 x + 6, express the width as a binomial. AE-44

19 Factor each difference of two squares. Additional Exercises 7.4 Form I Factoring Special Forms 1. x a y x 5 4. Factor any perfect square trinomials, or state that the polynomial is prime. 5. x + 10x y + 6y a 8a a 0a Factor each sum or difference of two cubes. 9. x AE-45

20 10. y x 8y a + 7b 1. Factor each polynomial completely, or state that the polynomial is prime. 13. x x 4x x 3 16x y + 0y x y x a 40a x + 48x AE-46

21 Factor each difference of two squares. Additional Exercises 7.4 Form II Factoring Special Forms 1. x a y x 5 4. Factor any perfect square trinomials, or state that the polynomial is prime. 5. x 4x y + 3y x + 10xy 5y x + 6xy Factor each sum or difference of two cubes. 9. x AE-47

22 a + 8b y x + y + 30xy 9 1. Factor each polynomial completely, or state that the polynomial is prime x x + 60x x ab 64a b y 3 9y x + 84x + 147x x x AE-48

23 Factor each difference of two squares. Additional Exercises 7.4 Form III Factoring Special Forms 1. 49x y x 1x a x 30x x + 48x a + 15b x + 14x x + y 3 36xy m 10. AE-49

24 11. 3x y x x + y + 100xy y 4 3 x x b 3 3 a m + n + 34mn x 81y ( x + 8) a + 150a AE-50

25 Additional Exercises 7.5 Form I A General Factoring Strategy Factor each polynomial completely, or state that the polynomial is prime. 1. x x + 10x a 14a x x + 17x x + 9x x x 1x xy + 5y 4x a + 4a y 3 5y 4y 10. AE-51

26 11. 6m + 17m x x 3 8x + 4x x 7x y b a x 5 3 x x y 9x y 1xy x x y + 16y 3x AE-5

27 Additional Exercises 7.5 Form II A General Factoring Strategy Factor each polynomial completely, or state that the polynomial is prime. 1. y x x 16x x y 4x + 6y 4 4, 5. 1x 5x y x a xy 3 x y x + 84x x + 41x AE-53

28 11. 48y xy 1x + 45y a a b + a b + ab x 64x x + 8x b 3 3 a x x + 1x + x x y 3xy + 4x 1x 0. AE-54

29 Additional Exercises 7.5 Form III A General Factoring Strategy Factor each polynomial completely, or state that the polynomial is prime. 1. a x + 9x y x y + 9x 8y a + 44a y x y + 14x y 4xy a b + 3a 50ab 40ab y 84y x AE-55

30 11. 45a 10a x + y + 0xy x y 5x y 50xy a y x 1x x y x + 16x a 8b b a AE-56

31 Date Additional Exercises 7.6 Form I Solving Quadratic Equations by Factoring Solve each equation using the zero product principle. 1. ( x 9)( x + 7) = ( y + 1)( y + 8) = ( 3x + 1)(x 1) = x ( x 5) = x ( x + 3)(5x 4) = 0 5. Use factoring to solve each quadratic equation. 6. x 5x + 4 = x 10x 8 = x x = x + 4x = 1 9. AE-57

32 Date 10. 6x = x y = x 7x + 60 = x 8x = x ( x 5) = y 30y = x = 1x Solve. 17. The width of a rectangle is 6 meters less than the length. 17. The area of the rectangle is 40 square meters. Find the dimensions of the rectangle. 18. An object is thrown upward from the top of a 160 foot 18. building with an initial velocity of 48 feet per second. The height h of the object after t seconds is given by the quadratic equation h = 16t + 48t How long will it take for the object to hit the ground? AE-58

33 Date Additional Exercises 7.6 Form II Solving Quadratic Equations by Factoring Solve each equation using the zero product principle. 1. ( x 7)( x 10) = ( y + 4)( y + 1) = ( 5x 4)( x + 9) = x ( x + 6)( x 7) = x ( x + 3)(4x 7) = 0 5. Use factoring to solve each quadratic equation. 6. y 9y + 8 = x + 13x 1 = x 3x = x 5 = 0 9. AE-59

34 Date 10. 3x 0x = x (4x + 13) = x = 14x y + 5y + 6 = x + 4x = x = x 8x x ( x = x + ) Solve. 17. The length of a rectangle is 8 inches less than twice the width. 17. If the area of the rectangle is 90 square inches, what are the dimensions of the rectangle? 18. A window washer accidentally drops a bucket from the top of a foot building. The height h of the bucket after t seconds is given by the quadratic equation h = 16t When will the bucket hit the ground? AE-60

35 Date Additional Exercises 7.6 Form III Solving Quadratic Equations by Factoring Solve each equation using the zero product principle. 1. ( 7x + 4)( x ) = ( x 8)(3x 10) = x ( 3x + 15) = y ( y + 17) = x ( 5x 1)( x 8) = 0 5. Use factoring to solve each quadratic equation. 6. 5x 4x 9 = x 4x = y 6y = x = 5 9. AE-61

36 Date 10. 4x = 0x x + 8x = 30x x ( 3x 5) = x ( 5x + 8) = x 3 75x = y( y + 5) = (36 y) x ( x 5) = 3( x + 4) 16. Solve. 17. The length of a rectangle is 6 feet less than twice the width. 17. If the area of the rectangle is 16 square feet, find the dimensions of the rectangle. 18. If the sides of a square are increased by meters, the area 18. becomes 64 square meters. Find the length of a side of the original square. 19. A window washer accidentally drops a bucket from the top of 19. a 100 foot tall building. The height h of the bucket after t seconds is given by h = 16t How long will it be before the bucket hits the ground? 0. Each cycle of a screen saver program generates and then erases 0. numbers of little animated figures called froobies. The formula P = x + 106x 674 models the population, P, of froobies after x minutes within a cycle. How many minutes into a cycle will the froobie population first reach 118? AE-6