Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Simplify each product.

9. 7 x ( x + 4 ) 7 x open x plus 4 close
10. ( b + 11 ) 2 b open b plus 11 close 2 b
11. 3 m 2 ( 10 + m ) 3 m squared , open 10 plus m close
12. − w 2 ( w − 15 ) negative , w squared , open w minus 15 close
13. 4 x ( 2 x 3 − 7 x 2 + x ) 4 x open , 2 x cubed , minus , 7 x squared , plus x close
14. − 8 y 3 ( 7 y 2 − 4 y − 1 ) negative , 8 y cubed , open , 7 y squared , minus 4 y minus 1 close

See Problem 2.

Find the GCF of the terms of each polynomial.

15. 12 x + 20 12 x plus 20
16. 8 w 2 − 18 w 8 w squared , minus 18 w
17. 45 b + 27 45 b plus 27
18. a 3 + 6 a 2 − 11 a eh cubed , plus , 6 eh squared , minus 11 eh
19. 4 x 3 + 12 x − 28 4 x cubed , plus 12 x minus 28
20. 14 z 4 − 42 z 3 + 21 z 2 14 z to the fourth , minus , 42 z cubed , plus , 21 z squared

See Problem 3.

Factor each polynomial.

21. 9 x − 6 9 x minus 6
22. t 2 + 8 t t squared , plus 8 t
23. 14 n 3 − 35 n 2 + 28 14 n cubed , minus , 35 n squared , plus 28
24. 5 k 3 + 20 k 2 − 15 5 k cubed , plus , 20 k squared , minus 15
25. 14 x 3 − 2 x 2 + 8 x 14 x cubed , minus , 2 x squared , plus 8 x
26. g 4 + 24 g 3 + 12 g 2 + 4 g g to the fourth , plus , 24 g cubed , plus , 12 g squared , plus 4 g

See Problem 4.

27. Art A circular mirror is surrounded by a square metal frame. The radius of the mirror is 5x. The side length of the metal frame is 15x. What is the area of the metal frame? Write your answer in factored form.

28. Design A circular table is painted yellow with a red square in the middle. The radius of the tabletop is 6x. The side length of the red square is 3x. What is the area of the yellow part of the tabletop? Write your answer in factored form.

    

B Apply

Simplify. Write in standard form.

29. − 2 x ( 5 x 2 − 4 x + 13 ) negative 2 x open , 5 x squared , minus 4 x plus 13 close
30. − 5 y 2 ( − 3 y 3 + 8 y ) negative , 5 y squared , open negative , 3 y cubed , plus 8 y close
31. 10 a ( − 6 a 2 + 2 a − 7 ) 10 eh open negative , 6 eh squared , plus 2 eh minus 7 close
32. p ( p + 2 ) − 3 p ( p − 5 ) p open p plus 2 close minus 3 p open p minus 5 close
33. t 2 ( t + 1 ) − t ( 2 t 2 − 1 ) t squared , open t plus 1 close minus t open , 2 t squared , minus 1 close
34. 3 c ( 4 c 2 − 5 ) − c ( 9 c ) 3 c open , 4 c squared , minus 5 close minus c open 9 c close

35. Think About a Plan A rectangular wooden frame has side lengths 5x and 7 x + 1 . 7 x plus 1 .

    The rectangular opening for a picture has side lengths 3x and 5x. What is the area of the wooden part of the frame? Write your answer in factored form.

    • How can drawing a diagram help you solve the problem?
    • How can you express the area of the wooden part of the frame as a difference of areas?

36. Error Analysis Describe and correct the error made in multiplying.

    

Factor each polynomial.

37. 17 x y 4 + 51 x 2 y 3 17 x , y to the fourth , plus , 51 x squared . y cubed
38. 9 m 4 n 5 − 27 m 2 n 3 9 m to the fourth . n to the fifth , minus , 27 m squared . n cubed
39. 31 a 6 b 3 + 63 a 5 31 eh to the sixth . b cubed , plus , 63 eh to the fifth
40. 1. Factor n 2 + n . n squared , plus n .
    2. Writing Suppose n is an integer. Is n 2 + n n squared , plus n  always, sometimes, or never an even integer? Justify your answer.

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