Section Exercises

1. If the terms of a polynomial do not have a GCF, does that mean it is not factorable? Explain.

2. A polynomial is factorable, but it is not a perfect square trinomial or a difference of two squares. Can you factor the polynomial without finding the GCF?

3. How do you factor by grouping?

For the following exercises, find the greatest common factor.

4. [latex]14x+4xy - 18x{y}^{2}[/latex]

5. [latex]49m{b}^{2}-35{m}^{2}ba+77m{a}^{2}[/latex]

6. [latex]30{x}^{3}y - 45{x}^{2}{y}^{2}+135x{y}^{3}\\[/latex]

7. [latex]200{p}^{3}{m}^{3}-30{p}^{2}{m}^{3}+40{m}^{3}\\[/latex]

8. [latex]36{j}^{4}{k}^{2}-18{j}^{3}{k}^{3}+54{j}^{2}{k}^{4}[/latex]

9. [latex]6{y}^{4}-2{y}^{3}+3{y}^{2}-y[/latex]

For the following exercises, factor by grouping.

10. [latex]6{x}^{2}+5x - 4[/latex]

11. [latex]2{a}^{2}+9a - 18[/latex]

12. [latex]6{c}^{2}+41c+63[/latex]

13. [latex]6{n}^{2}-19n - 11[/latex]

14. [latex]20{w}^{2}-47w+24[/latex]

15. [latex]2{p}^{2}-5p - 7[/latex]

For the following exercises, factor the polynomial.

16. [latex]7{x}^{2}+48x - 7[/latex]

17. [latex]10{h}^{2}-9h - 9[/latex]

18. [latex]2{b}^{2}-25b - 247[/latex]

19. [latex]9{d}^{2}-73d+8[/latex]

20. [latex]90{v}^{2}-181v+90[/latex]

21. [latex]12{t}^{2}+t - 13[/latex]

22. [latex]2{n}^{2}-n - 15[/latex]

23. [latex]16{x}^{2}-100[/latex]

24. [latex]25{y}^{2}-196[/latex]

25. [latex]121{p}^{2}-169[/latex]

26. [latex]4{m}^{2}-9[/latex]

27. [latex]361{d}^{2}-81[/latex]

28. [latex]324{x}^{2}-121[/latex]

29. [latex]144{b}^{2}-25{c}^{2}[/latex]

30. [latex]16{a}^{2}-8a+1[/latex]

31. [latex]49{n}^{2}+168n+144[/latex]

32. [latex]121{x}^{2}-88x+16[/latex]

33. [latex]225{y}^{2}+120y+16[/latex]

34. [latex]{m}^{2}-20m+100[/latex]

35. [latex]{m}^{2}-20m+100[/latex]

36. [latex]36{q}^{2}+60q+25[/latex]

For the following exercises, factor the polynomials.

37. [latex]{x}^{3}+216[/latex]

38. [latex]27{y}^{3}-8[/latex]

39. [latex]125{a}^{3}+343[/latex]

40. [latex]{b}^{3}-8{d}^{3}[/latex]

41. [latex]64{x}^{3}-125[/latex]

42. [latex]729{q}^{3}+1331[/latex]

43. [latex]125{r}^{3}+1,728{s}^{3}[/latex]

44. [latex]4x{\left(x - 1\right)}^{-\frac{2}{3}}+3{\left(x - 1\right)}^{\frac{1}{3}}[/latex]

45. [latex]3c{\left(2c+3\right)}^{-\frac{1}{4}}-5{\left(2c+3\right)}^{\frac{3}{4}}[/latex]

46. [latex]3t{\left(10t+3\right)}^{\frac{1}{3}}+7{\left(10t+3\right)}^{\frac{4}{3}}[/latex]

47. [latex]14x{\left(x+2\right)}^{-\frac{2}{5}}+5{\left(x+2\right)}^{\frac{3}{5}}[/latex]

48. [latex]9y{\left(3y - 13\right)}^{\frac{1}{5}}-2{\left(3y - 13\right)}^{\frac{6}{5}}[/latex]

49. [latex]5z{\left(2z - 9\right)}^{-\frac{3}{2}}+11{\left(2z - 9\right)}^{-\frac{1}{2}}[/latex]

50. [latex]6d{\left(2d+3\right)}^{-\frac{1}{6}}+5{\left(2d+3\right)}^{\frac{5}{6}}[/latex]

For the following exercises, consider this scenario:
Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area of [latex]98{x}^{2}+105x - 27[/latex] m2, as shown in the figure below. The length and width of the park are perfect factors of the area.

A rectangle that’s textured to look like a field. The field is labeled: l times w = ninety-eight times x squared plus one hundred five times x minus twenty-seven.
51. Factor by grouping to find the length and width of the park.

52. A statue is to be placed in the center of the park. The area of the base of the statue is [latex]4{x}^{2}+12x+9{\text{m}}^{2}[/latex]. Factor the area to find the lengths of the sides of the statue.

53. At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is [latex]9{x}^{2}-25{\text{m}}^{2}[/latex]. Factor the area to find the lengths of the sides of the fountain.

For the following exercise, consider the following scenario:
A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below. The flagpole will take up a square plot with area [latex]{x}^{2}-6x+9[/latex] yd2.

A square that’s textured to look like a field with a missing piece in the shape of a square in the center. The sides of the larger square are labeled: 100 yards. The center square is labeled: Area: x squared minus six times x plus nine.
54. Find the length of the base of the flagpole by factoring.

For the following exercises, factor the polynomials completely.

55. [latex]16{x}^{4}-200{x}^{2}+625[/latex]

56. [latex]81{y}^{4}-256[/latex]

57. [latex]16{z}^{4}-2,401{a}^{4}[/latex]

58. [latex]5x{\left(3x+2\right)}^{-\frac{2}{4}}+{\left(12x+8\right)}^{\frac{3}{2}}[/latex]

59. [latex]{\left(32{x}^{3}+48{x}^{2}-162x - 243\right)}^{-1}[/latex]