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# 5.4E: Exercises


## Practice Makes Perfect

Divide Monomials

In the following exercises, divide the monomials.

1. $$15r^4s^9÷(15r^4s^9)$$

2. $$20m^8n^4÷(30m^5n^9)$$

Answer

$$\dfrac{2m^3}{3n^5}$$

3. $$\dfrac{18a^4b^8}{−27a^9b^5}$$

4. $$\dfrac{45x^5y^9}{−60x^8y^6}$$

Answer

$$\dfrac{−3y^3}{4x^3}$$

5. $$\dfrac{(10m^5n^4)(5m^3n^6)}{25m^7n^5}$$

6. $$\dfrac{(−18p^4q^7)(−6p^3q^8)}{−36p^{12}q^{10}}$$

Answer

$$\dfrac{−3q^5}{p^5}$$

7. $$\dfrac{(6a^4b^3)(4ab^5)}{(12a^2b)(a^3b)}$$

8. $$\dfrac{(4u^2v^5)(15u^3v)}{(12u^3v)(u^4v)}$$

Answer

$$\dfrac{5v^4}{u^2}$$

Divide a Polynomial by a Monomial

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

9. $$(9n^4+6n^3)÷3n$$

10. $$(8x^3+6x^2)÷2x$$

Answer

$$4x^2+3x$$

11. $$(63m^4−42m^3)÷(−7m^2)$$

12. $$(48y^4−24y^3)÷(−8y^2)$$

Answer

$$−6y^2+3y$$

13. $$\dfrac{66x^3y^2−110x^2y^3−44x^4y^3}{11x^2y^2}$$

14. $$\dfrac{72r^5s^2+132r^4s^3−96r^3s^5}{12r^2s^2}$$

Answer

$$6r^3+11r^2s−8rs^3$$

15. $$10x^2+5x−4−5x$$

16. $$20y^2+12y−1−4y$$

Answer

$$−5y−3+\dfrac{1}{4y}$$

Divide Polynomials using Long Division

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

17. $$(y^2+7y+12)÷(y+3)$$

18. $$(a^2−2a−35)÷(a+5)$$

Answer

$$a−7$$

19. $$(6m^2−19m−20)÷(m−4)$$

20. $$(4x^2−17x−15)÷(x−5)$$

Answer

$$4x+3$$

21. $$(q^2+2q+20)÷(q+6)$$

22. $$(p^2+11p+16)÷(p+8)$$

Answer

$$p+3−\dfrac{8}{p+8}$$

23. $$(3b^3+b^2+4)÷(b+1)$$

24. $$(2n^3−10n+28)÷(n+3)$$

Answer

$$\dfrac{2n^2−6n+8+4}{n+3}$$

25. $$(z^3+1)÷(z+1)$$

26. $$(m^3+1000)÷(m+10)$$

Answer

$$m^2−10m+100$$

27. $$(64x^3−27)÷(4x−3)$$

28. $$(125y^3−64)÷(5y−4)$$

Answer

$$25y^2+20x+16$$

Divide Polynomials using Synthetic Division

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

29. $$x^3−6x^2+5x+14$$ is divided by $$x+1$$

30. $$x^3−3x^2−4x+12$$ is divided by $$x+2$$

Answer

$$x^2−5x+6; \space 0$$

31. $$2x^3−11x^2+11x+12$$ is divided by $$x−3$$

32. $$2x^3−11x^2+16x−12$$ is divided by $$x−4$$

Answer

$$2x^2−3x+4; \space 4$$

33. $$x^4-5x^2+2+13x+3$$ is divided by $$x+3$$

34. $$x^4+x^2+6x−10$$ is divided by $$x+2$$

Answer

$$x^3−2x^2+5x−4; \space −2$$

35. $$2x^4−9x^3+5x^2−3x−6$$ is divided by $$x−4$$

36. $$3x^4−11x^3+2x^2+10x+6$$ is divided by $$x−3$$

Answer

$$3x^3−2x^2−4x−2;\space 0$$

Divide Polynomial Functions

In the following exercises, divide.

37. For functions $$f(x)=x^2−13x+36$$ and $$g(x)=x−4$$, find ⓐ $$\left(\dfrac{f}{g}\right)(x)$$ ⓑ $$\left(\dfrac{f}{g}\right)(−1)$$

38. For functions $$f(x)=x^2−15x+54$$ and $$g(x)=x−9$$, find ⓐ $$\left(\dfrac{f}{g}\right)(x)$$ ⓑ $$\left(\dfrac{f}{g}\right)(−5)$$

Answer

ⓐ $$\left(\dfrac{f}{g}\right)(x)=x−6$$
ⓑ $$\left(\dfrac{f}{g}\right)(−5)=−11$$

39. For functions $$f(x)=x^3+x^2−7x+2$$ and $$g(x)=x−2$$, find ⓐ $$\left(\dfrac{f}{g}\right)(x)$$ ⓑ $$\left(\dfrac{f}{g}\right)(2)$$

40. For functions $$f(x)=x^3+2x^2−19x+12$$ and $$g(x)=x−3$$, find ⓐ $$\left(\dfrac{f}{g}\right)(x)$$ ⓑ $$\left(\dfrac{f}{g}\right)(0)$$

Answer

ⓐ $$\left(\dfrac{f}{g}\right)(x)=x^2+5x−4$$
ⓑ $$\left(\dfrac{f}{g}\right)(0)=−4$$

41. For functions $$f(x)=x^2−5x+2$$ and $$g(x)=x^2−3x−1$$, find ⓐ $$(f·g)(x)$$ ⓑ $$(f·g)(−1)$$

42. For functions $$f(x)=x^2+4x−3$$ and $$g(x)=x^2+2x+4$$, find ⓐ $$(f·g)(x)$$ ⓑ $$(f·g)(1)$$

Answer

ⓐ $$(f·g)(x)=x^4+6x^3+9x^2+10x−12$$; ⓑ $$(f·g)(1)=14$$

Use the Remainder and Factor Theorem

In the following exercises, use the Remainder Theorem to find the remainder.

43. $$f(x)=x^3−8x+7$$ is divided by $$x+3$$

44. $$f(x)=x^3−4x−9$$ is divided by $$x+2$$

Answer

$$−9$$

45. $$f(x)=2x^3−6x−24$$ divided by $$x−3$$

46. $$f(x)=7x^2−5x−8$$ 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 $$x^3+8x^2+21x+18$$

48. Determine whether $$x+4$$ a factor of $$x^3+x^2−14x+8$$

Answer

no

49. Determine whether $$x−2$$ a factor of $$x^3−7x^2+7x−6$$

50. Determine whether $$x−3$$ a factor of $$x^3−7x^2+11x+3$$

Answer

yes

## Writing Exercises

51. James divides $$48y+6$$ by $$6$$ this way: $$\dfrac{48y+6}{6}=48y$$. What is wrong with his reasoning?

52. Divide $$\dfrac{10x^2+x−12}{2x}$$ 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 $$x^2+5x+6$$ 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?

This page titled 5.4E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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