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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. 18a4b827a9b5

4. 45x5y960x8y6

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. (63m442m3)÷(7m2)

12. (48y424y3)÷(8y2)

Answer

6y2+3y

13. 66x3y2110x2y344x4y311x2y2

14. 72r5s2+132r4s396r3s512r2s2

Answer

6r3+11r2s8rs3

15. 10x2+5x45x

16. 20y2+12y14y

Answer

5y3+14y

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

17. (y2+7y+12)÷(y+3)

18. (a22a35)÷(a+5)

Answer

a7

19. (6m219m20)÷(m4)

20. (4x217x15)÷(x5)

Answer

4x+3

21. (q2+2q+20)÷(q+6)

22. (p2+11p+16)÷(p+8)

Answer

p+38p+8

23. (3b3+b2+4)÷(b+1)

24. (2n310n+28)÷(n+3)

Answer

2n26n+8+4n+3

25. (z3+1)÷(z+1)

26. (m3+1000)÷(m+10)

Answer

m210m+100

27. (64x327)÷(4x3)

28. (125y364)÷(5y4)

Answer

25y2+20x+16

Divide Polynomials using Synthetic Division

In the following exercises, use synthetic Division to find the quotient and remainder.

29. x36x2+5x+14 is divided by x+1

30. x33x24x+12 is divided by x+2

Answer

x25x+6; 0

31. 2x311x2+11x+12 is divided by x3

32. 2x311x2+16x12 is divided by x4

Answer

2x23x+4; 4

33. x45x2+2+13x+3 is divided by x+3

34. x4+x2+6x10 is divided by x+2

Answer

x32x2+5x4; 2

35. 2x49x3+5x23x6 is divided by x4

36. 3x411x3+2x2+10x+6 is divided by x3

Answer

3x32x24x2; 0

Divide Polynomial Functions

In the following exercises, divide.

37. For functions f(x)=x213x+36 and g(x)=x4, find ⓐ (fg)(x)(fg)(1)

38. For functions f(x)=x215x+54 and g(x)=x9, find ⓐ (fg)(x)(fg)(5)

Answer

(fg)(x)=x6
(fg)(5)=11

39. For functions f(x)=x3+x27x+2 and g(x)=x2, find ⓐ (fg)(x)(fg)(2)

40. For functions f(x)=x3+2x219x+12 and g(x)=x3, find ⓐ (fg)(x)(fg)(0)

Answer

(fg)(x)=x2+5x4
(fg)(0)=4

41. For functions f(x)=x25x+2 and g(x)=x23x1, find ⓐ (f·g)(x)(f·g)(1)

42. For functions f(x)=x2+4x3 and g(x)=x2+2x+4, find ⓐ (f·g)(x)(f·g)(1)

Answer

(f·g)(x)=x4+6x3+9x2+10x12; ⓑ (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)=x38x+7 is divided by x+3

44. f(x)=x34x9 is divided by x+2

Answer

9

45. f(x)=2x36x24 divided by x3

46. f(x)=7x25x8 divided by x1

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+x214x+8

Answer

no

49. Determine whether x2 a factor of x37x2+7x6

50. Determine whether x3 a factor of x37x2+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+x122x 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 x2.

Answer

Answers will vary.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section

The figure shows a table with seven rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “divide monomials”, “divide a polynomial by using a monomial”, “divide polynomials using long division”, “divide polynomials using synthetic division”, “divide polynomial functions”, and “use the Remainder and Factor Theorem”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

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?


This page titled 5.5E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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