Section Exercises
1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?
2. If a polynomial of degree n is divided by a binomial of degree 1, what is the degree of the quotient?
For the following exercises, use long division to divide. Specify the quotient and the remainder.
3. (x2+5x−1)÷(x−1)
4. (2x2−9x−5)÷(x−5)
5. (3x2+23x+14)÷(x+7)
6. (4x2−10x+6)÷(4x+2)
7. (6x2−25x−25)÷(6x+5)
8. (−x2−1)÷(x+1)
9. (2x2−3x+2)÷(x+2)
10. (x3−126)÷(x−5)
11. (3x2−5x+4)÷(3x+1)
12. (x3−3x2+5x−6)÷(x−2)
13. (2x3+3x2−4x+15)÷(x+3)
For the following exercises, use synthetic division to find the quotient.
14. (3x3−2x2+x−4)÷(x+3)
15. (2x3−6x2−7x+6)÷(x−4)
16. (6x3−10x2−7x−15)÷(x+1)
17. (4x3−12x2−5x−1)÷(2x+1)
18. (9x3−9x2+18x+5)÷(3x−1)
19. (3x3−2x2+x−4)÷(x+3)
20. (−6x3+x2−4)÷(2x−3)
21. (2x3+7x2−13x−3)÷(2x−3)
22. (3x3−5x2+2x+3)÷(x+2)
23. (4x3−5x2+13)÷(x+4)
24. (x3−3x+2)÷(x+2)
25. (x3−21x2+147x−343)÷(x−7)
26. (x3−15x2+75x−125)÷(x−5)
27. (9x3−x+2)÷(3x−1)
28. (6x3−x2+5x+2)÷(3x+1)
29. (x4+x3−3x2−2x+1)÷(x+1)
30. (x4−3x2+1)÷(x−1)
31. (x4+2x3−3x2+2x+6)÷(x+3)
32. (x4−10x3+37x2−60x+36)÷(x−2)
33. (x4−8x3+24x2−32x+16)÷(x−2)
34. (x4+5x3−3x2−13x+10)÷(x+5)
35. (x4−12x3+54x2−108x+81)÷(x−3)
36. (4x4−2x3−4x+2)÷(2x−1)
37. (4x4+2x3−4x2+2x+2)÷(2x+1)
For the following exercises, use the graph of the third-degree polynomial and one factor to write the factored form of the polynomial suggested by the graph. The leading coefficient is one.
38. Factor is x2−x+3
39. Factor is (x2+2x+4)
40. Factor is x2+2x+5
41. Factor is x2+x+1
42. Factor is x2+2x+2
For the following exercises, use synthetic division to find the quotient and remainder.
43. x−24x3−33
44. x+32x3+25
45. x−13x3+2x−5
46. x+4−4x3−x2−12
47. x+2x4−22
For the following exercises, use a calculator with CAS to answer the questions.
48. Consider x−1xk−1 with k=1,2,3. What do you expect the result to be if k = 4?
49. Consider x+1xk+1 for k=1,3,5. What do you expect the result to be if k = 7?
50. Consider x−kx4−k4 for k=1,2,3. What do you expect the result to be if k = 4?
51. Consider x+1xk with k=1,2,3. What do you expect the result to be if k = 4?
52. Consider x−1xk with k=1,2,3. What do you expect the result to be if k = 4?
For the following exercises, use synthetic division to determine the quotient involving a complex number.
53. x−ix+1
54. x−ix2+1
55. x+ix+1
56. x+ix2+1
57. x−ix3+1
For the following exercises, use the given length and area of a rectangle to express the width algebraically.
58. Length is x+5, area is 2x2+9x−5.
59. Length is 2x + 5, area is 4x3+10x2+6x+15
60. Length is 3x−4, area is 6x4−8x3+9x2−9x−4
For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.
61. Volume is 12x3+20x2−21x−36, length is 2x+3, width is 3x−4.
62. Volume is 18x3−21x2−40x+48, length is 3x−4, width is 3x−4.
63. Volume is 10x3+27x2+2x−24, length is 5x−4, width is 2x+3.
64. Volume is 10x3+30x2−8x−24, length is 2, width is x+3.
For the following exercises, use the given volume and radius of a cylinder to express the height of the cylinder algebraically.
65. Volume is π(25x3−65x2−29x−3), radius is 5x+1.
66. Volume is π(4x3+12x2−15x−50), radius is 2x+5.
67. Volume is π(3x4+24x3+46x2−16x−32), radius is x+4.
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175..