5.5E: Exercises
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Practice Makes Perfect
Divide Monomials
In the following exercises, divide the monomials.
1. 15r4s9÷(15r4s9)
2. 20m8n4÷(30m5n9)
- Answer
-
2m33n5
3. 18a4b8−27a9b5
4. 45x5y9−60x8y6
- Answer
-
−3y34x3
5. (10m5n4)(5m3n6)25m7n5
6. (−18p4q7)(−6p3q8)−36p12q10
- Answer
-
−3q5p5
7. (6a4b3)(4ab5)(12a2b)(a3b)
8. (4u2v5)(15u3v)(12u3v)(u4v)
- Answer
-
5v4u2
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
9. (9n4+6n3)÷3n
10. (8x3+6x2)÷2x
- Answer
-
4x2+3x
11. (63m4−42m3)÷(−7m2)
12. (48y4−24y3)÷(−8y2)
- Answer
-
−6y2+3y
13. 66x3y2−110x2y3−44x4y311x2y2
14. 72r5s2+132r4s3−96r3s512r2s2
- Answer
-
6r3+11r2s−8rs3
15. 10x2+5x−4−5x
16. 20y2+12y−1−4y
- Answer
-
−5y−3+14y
Divide Polynomials using Long Division
In the following exercises, divide each polynomial by the binomial.
17. (y2+7y+12)÷(y+3)
18. (a2−2a−35)÷(a+5)
- Answer
-
a−7
19. (6m2−19m−20)÷(m−4)
20. (4x2−17x−15)÷(x−5)
- Answer
-
4x+3
21. (q2+2q+20)÷(q+6)
22. (p2+11p+16)÷(p+8)
- Answer
-
p+3−8p+8
23. (3b3+b2+4)÷(b+1)
24. (2n3−10n+28)÷(n+3)
- Answer
-
2n2−6n+8+4n+3
25. (z3+1)÷(z+1)
26. (m3+1000)÷(m+10)
- Answer
-
m2−10m+100
27. (64x3−27)÷(4x−3)
28. (125y3−64)÷(5y−4)
- Answer
-
25y2+20x+16
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
29. x3−6x2+5x+14 is divided by x+1
30. x3−3x2−4x+12 is divided by x+2
- Answer
-
x2−5x+6; 0
31. 2x3−11x2+11x+12 is divided by x−3
32. 2x3−11x2+16x−12 is divided by x−4
- Answer
-
2x2−3x+4; 4
33. x4−5x2+2+13x+3 is divided by x+3
34. x4+x2+6x−10 is divided by x+2
- Answer
-
x3−2x2+5x−4; −2
35. 2x4−9x3+5x2−3x−6 is divided by x−4
36. 3x4−11x3+2x2+10x+6 is divided by x−3
- Answer
-
3x3−2x2−4x−2; 0
Divide Polynomial Functions
In the following exercises, divide.
37. For functions f(x)=x2−13x+36 and g(x)=x−4, find ⓐ (fg)(x) ⓑ (fg)(−1)
38. For functions f(x)=x2−15x+54 and g(x)=x−9, find ⓐ (fg)(x) ⓑ (fg)(−5)
- Answer
-
ⓐ (fg)(x)=x−6
ⓑ (fg)(−5)=−11
39. For functions f(x)=x3+x2−7x+2 and g(x)=x−2, find ⓐ (fg)(x) ⓑ (fg)(2)
40. For functions f(x)=x3+2x2−19x+12 and g(x)=x−3, find ⓐ (fg)(x) ⓑ (fg)(0)
- Answer
-
ⓐ (fg)(x)=x2+5x−4
ⓑ (fg)(0)=−4
41. For functions f(x)=x2−5x+2 and g(x)=x2−3x−1, find ⓐ (f·g)(x) ⓑ (f·g)(−1)
42. For functions f(x)=x2+4x−3 and g(x)=x2+2x+4, find ⓐ (f·g)(x) ⓑ (f·g)(1)
- Answer
-
ⓐ (f·g)(x)=x4+6x3+9x2+10x−12; ⓑ (f·g)(1)=14
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
43. f(x)=x3−8x+7 is divided by x+3
44. f(x)=x3−4x−9 is divided by x+2
- Answer
-
−9
45. f(x)=2x3−6x−24 divided by x−3
46. f(x)=7x2−5x−8 divided by x−1
- Answer
-
−6
In the following exercises, use the Factor Theorem to determine if x−cx−c is a factor of the polynomial function.
47. Determine whether x+3 a factor of x3+8x2+21x+18
48. Determine whether x+4 a factor of x3+x2−14x+8
- Answer
-
no
49. Determine whether x−2 a factor of x3−7x2+7x−6
50. Determine whether x−3 a factor of x3−7x2+11x+3
- Answer
-
yes
Writing Exercises
51. James divides 48y+6 by 6 this way: 48y+66=48y. What is wrong with his reasoning?
52. Divide 10x2+x−122x and explain with words how you get each term of the quotient.
- Answer
-
Answer will vary
53. Explain when you can use synthetic division.
54. In your own words, write the steps for synthetic division for x2+5x+6 divided by x−2.
- Answer
-
Answers will vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section

b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?