8.7 Chapter Review

Izabela Mazur

CHAPTER 8 Polynomials

8.7 Chapter Review

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

1.

a) ${a}^{2}-{b}^{2}$
b) $24{d}^{3}$
c) ${x}^{2}+8x-10$
d) ${m}^{2}{n}^{2}-2mn+6$
e) $7{y}^{3}+{y}^{2}-2y-4$

2.

a) $11{c}^{4}-23{c}^{2}+1$
b) $9{p}^{3}+6{p}^{2}-p-5$
c) $\frac{3}{7}x+\frac{5}{14}$
d) 10
e) $2y-12$

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

3.

a) $5{p}^{3}-8{p}^{2}+10p-4$

b) $-20{q}^{4}$

c) ${x}^{2}+6x+12$

d) $23{r}^{2}{s}^{2}-4rs+5$

e) 100

4.

a) $3{x}^{2}+9x+10$

b) $14{a}^{2}bc$

c) $6y+1$

d) ${n}^{3}-4{n}^{2}+2n-8$

e) $-19$

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

5. $-14k+19k$	6. ${\phantom{\rule{0.2em}{0ex}}\text{5y}}^{\text{3}}+8{y}^{3}$
7. $-9c-18c$	8. $12q-\left(-6q\right)$
9. $3{m}^{2}+7{n}^{2}-3{m}^{2}$	10. $\text{12x}-4y-9x$
11. $\text{13a}+b$	12. $6{x}^{2}y-4x+8x{y}^{2}$

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

13. $\left(9{p}^{2}-5p+3\right)+\left(4{p}^{2}-4\right)$	14. $\left(5{x}^{2}+12x+1\right)+\left(6{x}^{2}-8x+3\right)$
15. $\left(7{y}^{2}-8y\right)-\left(y-4\right)$	16. $\left(10{m}^{2}-8m-1\right)-\left(5{m}^{2}+m-2\right)$
17. Find the sum of $\left({a}^{2}+6a+9\right)\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(5{a}^{3}-7\right)$	18. Subtract $\left(3{s}^{2}+10\right)\phantom{\rule{0.2em}{0ex}}\text{from}\phantom{\rule{0.2em}{0ex}}\left(15{s}^{2}-2s+8\right)$

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

19. Evaluate $10-12x$ when:

a) $x=3$

b) $x=0$

c) $x=-1$

20. Evaluate $3{y}^{2}-y+1$ when:

a) $y=5$

b) $y=-1$

c) $y=0$

21. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $-4{p}^{2}+460p$ . Find the revenue received when $p=75$ dollars.

22. Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial $-16{t}^{2}+200$ gives the height of a stone $t$ seconds after it is dropped from the cliff. Find the height after $t=3$ seconds.

Multiply Monomials

In the following exercises, multiply the monomials.

23. $\left(-9{n}^{7}\right)\left(-16n\right)$	24. $\left(-15{x}^{2}\right)\left(6{x}^{4}\right)$
25. $\left(\frac{5}{9}a{b}^{2}\right)\left(27a{b}^{3}\right)$	26. $\left(7{p}^{5}{q}^{3}\right)\left(8p{q}^{9}\right)$

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

27. $-4\left(y+13\right)$	28. $7\left(a+9\right)$
29. $p\left(p+3\right)$	30. $-5\left(r-2\right)$
31. $-6u\left(2u+7\right)$	32. $-m\left(m+15\right)$
33. $3{q}^{2}\left({q}^{2}-7q+6\right)$ 3	34. $9\left({b}^{2}+6b+8\right)$
35. $\left(b-4\right)\cdot 11$	36. $\left(5z-1\right)z$

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: a) the Distributive Property, b) the FOIL method, c) the Vertical Method.

37. $\left(6y-7\right)\left(2y-5\right)$

38. $\left(x-4\right)\left(x+10\right)$

In the following exercises, multiply the binomials. Use any method.

39. $\left(y-4\right)\left(y-8\right)$	40. $\left(x+3\right)\left(x+9\right)$
41. $\left(q+16\right)\left(q-3\right)$	42. $\left(p-7\right)\left(p+4\right)$
43. $\left({u}^{2}+6\right)\left({u}^{2}-5\right)$	44. $\left(5m-8\right)\left(12m+1\right)$
45. $\left(8mn+3\right)\left(2mn-1\right)$	46. $\left(9x-y\right)\left(6x-5\right)$

Multiply a Trinomial by a Binomial

In the following exercises, multiply using a) the Distributive Property, b) the Vertical Method.

47. $\left(3x-4\right)\left(6{x}^{2}+x-10\right)$

48. $\left(n+1\right)\left({n}^{2}+5n-2\right)$

In the following exercises, multiply. Use either method.

49. $\left(7m+1\right)\left({m}^{2}-10m-3\right)$

50. $\left(y-2\right)\left({y}^{2}-8y+9\right)$

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

51. ${\left(q-15\right)}^{2}$	52. ${\left(c+11\right)}^{2}$
53. ${\left(8u+1\right)}^{2}$	54. ${\left(x+\frac{1}{3}\right)}^{2}$
55. ${\left(4a-3b\right)}^{2}$	56. ${\left(3{n}^{3}-2\right)}^{2}$

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

57. $\left(y+\frac{2}{5}\right)\left(y-\frac{2}{5}\right)$	58. $\left(s-7\right)\left(s+7\right)$
59. $\left(6-r\right)\left(6+r\right)$	60. $\left(12c+13\right)\left(12c-13\right)$
61. $\left(5{p}^{4}-4{q}^{3}\right)\left(5{p}^{4}+4{q}^{3}\right)$	62. $\left(u+\frac{3}{4}v\right)\left(u-\frac{3}{4}v\right)$

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

63. $\left(6a+11\right)\left(6a-11\right)$	64. ${\left(3m+10\right)}^{2}$
65. ${\left({c}^{4}+9d\right)}^{2}$	66. $\left(5x+y\right)\left(x-5y\right)$
67. $\left({a}^{2}+4b\right)\left(4a-{b}^{2}\right)$	68. $\left({p}^{5}+{q}^{5}\right)\left({p}^{5}-{q}^{5}\right)$

Divide a Polynomial by a Monomial

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

69. $\left(35{x}^{2}-75x\right)\div5x$	70. $\frac{42{z}^{2}-18z}{6}$
71. $\frac{550{p}^{6}-300{p}^{4}}{10{p}^{3}}$	72. $\frac{81{n}^{4}+105{n}^{2}}{-3}$
73. $\frac{96{a}^{5}{b}^{2}-48{a}^{4}{b}^{3}-56{a}^{2}{b}^{4}}{8a{b}^{2}}$	74. $\left(63x{y}^{3}+56{x}^{2}{y}^{4}\right)\div\left(7xy\right)$
75. $\frac{105{y}^{5}+50{y}^{3}-5y}{5{y}^{3}}$	76. $\frac{57{m}^{2}-12m+1}{-3m}$

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1. a) binomial b) monomial c) trinomial d) trinomial e) other polynomial	3. a) 3 b) 4 c) 2 d) 4 e) 0	5. $5k$
7. $-27c$	9. $7{n}^{2}$	11. $\text{13a}+b$
13. $13{p}^{2}-5p-1$	15. $7{y}^{2}-9y+4$	17. $5{a}^{3}+{a}^{2}+6a+2$
19. a) $-26$ b) 10 c) 22	21. 12,000	23. $144{n}^{8}$
25. $15{a}^{2}{b}^{5}$	27. $-4y-52$	29. ${p}^{2}+3p$
31. $-12{u}^{2}-42u$	33. $3{q}^{4}-21{q}^{3}+18{q}^{2}$	35. $11b-44$
37. a) $12{y}^{2}-44y+35$ b) $12{y}^{2}-44y+35$ c) $12{y}^{2}-44y+35$	39. ${y}^{2}-12y+32$	41. ${q}^{2}+13q-48$
43. ${u}^{4}+{u}^{2}-30$	45. $16{m}^{2}{n}^{2}-2mn-3$	47. a) $18{x}^{3}-21{x}^{2}-34x+40$ b) $18{x}^{3}-21{x}^{2}-34x+40$
49. $7{m}^{3}-69{m}^{2}-31m-3$	51. ${q}^{2}-30q+225$	53. $64{u}^{2}+16u+1$
55. $16{a}^{2}-24ab+9{b}^{2}$	57. ${y}^{2}-\frac{4}{25}$	59. $36-{r}^{2}$
61. $25{p}^{8}-16{q}^{6}$	63. $36{a}^{2}-121$	65. ${c}^{8}+18{c}^{4}d+81{d}^{2}$
67. $4{a}^{3}+3{a}^{2}b-4{b}^{3}$	69. $7x-15$	71. $55{p}^{3}-30p$
73. $12{a}^{4}-6{a}^{3}b-7a{b}^{2}$	75. $21{y}^{2}+10-\frac{1}{{y}^{2}}$

In the following exercises, simplify each expression. 1. $\left(12{a}^{2}-7a+4\right)+\left(3{a}^{2}+8a-10\right)$	2. For the polynomial $10{x}^{4}+9{y}^{2}-1$ a) Is it a monomial, binomial, or trinomial? b) What is its degree?
3. $\left(9{p}^{2}-5p+1\right)-\left(2{p}^{2}-6\right)$	4. $\left(-9{r}^{4}{s}^{5}\right)\left(4r{s}^{7}\right)$
5. $\left(v-9\right)\left(9v-5\right)$	6. $\left(m+6\right)\left(m+12\right)$
7. $\left(n-6\right)\left({n}^{2}-5n+4\right)$	8. $\left(4c-11\right)\left(3c-8\right)$
9. $\left(7p-5\right)\left(7p+5\right)$	10. $\left(2x-15y\right)\left(5x+7y\right)$
11. ${\left(9v-2\right)}^{2}$	12. $\frac{12{x}^{3}+42{x}^{2}-6x}{2x}$
13. $\frac{64{x}^{3}-x}{4x}$	14. $\frac{70x{y}^{4}+95{x}^{3}y}{5xy}$
15. $\frac{{y}^{2}-5y-18}{y}$	16. A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $-16{t}^{2}+1000$ gives the height of the package $t$ seconds after it was dropped. Find the height when $t=6$ seconds.

1. $15{a}^{2}+a-6$	2. a) Trinomial, b) 4	3. $7{p}^{2}-5p+7$
4. $-36{r}^{5}{s}^{12}$	5. $9{v}^{2}-86v+45$	6. ${m}^{2}+18m+72$
7. ${n}^{3}-11{n}^{2}+34n-24$	8. $12{c}^{2}-65c+88$	9. $49{p}^{2}-25$
10. $10{x}^{2}-61xy-105{y}^{2}$	11. $81{v}^{2}-36v+4$	12. $6{x}^{2}+21x-3$
13. $16{x}^{2}-\frac{1}{4}$	14. $14 y^{3}+19{x}^{2}$	15. $y -5-$ $\frac{18}{y}$
16. 424 feet

Review Exercises