ⓐ FOIL; $18 b^{2} + 77 b - 18$
ⓑ Binomial Squares; $81 p^{2} - 72 p + 16$
ⓒ Binomial Squares; $49 y^{2} + 14 y + 1$
ⓓ Product of Conjugates; $16 r^{2} - 9$

5.68

ⓐ Binomial Squares; $36 x^{2} + 84 x + 49$ ⓑ Product of Conjugates; $9 x^{2} - 16$ ⓒ FOIL; $10 x^{2} - 29 x + 10$ ⓓ Binomial Squares; $36 n^{2} - 12 n + 1$

5.69

ⓐ $(f \cdot g) (x) = x^{3} - 7 x^{2} + 13 x - 15$
ⓑ $(f \cdot g) (2) = −9$

5.70

ⓐ $(f \cdot g) (x) = x^{3} + x^{2} - 52 x - 28$
ⓑ $(f \cdot g) (2) = −120$

5.71

$- \frac{9}{a^{5} b}$

5.72

$\frac{−9 d}{c^{4}}$

5.73

$\frac{4 y^{2}}{7 x^{4}}$

5.74

$\frac{5}{8 m^{5} n^{3}}$

5.75

$−4 a + 2 b$

5.76

$8 a^{5} b + 6 a^{3} b^{2}$

5.77

$y + 7$

5.78

$m + 5$

5.79

$x^{3} - 3 x^{2} + 2 x + 1 + \frac{3}{x + 3}$

5.80

$x^{3} - 3 x^{2} - 2 x - 1 - \frac{3}{x + 3}$

5.81

$x^{2} + 4 x + 16$

5.82

$25 x^{2} + 10 x + 4$

5.83

$3 x^{2} + 4 x - 2; 2$

5.84

$4 x^{2} - 3 x + 1; 1$

5.85

$x^{3} - 4 x^{2} + 5; 0$

5.86

$x^{3} - 3 x^{2} + 2; 0$

5.87

ⓐ $(\frac{f}{g}) (x) = x - 8$
ⓑ $(\frac{f}{g}) (−3) = −11$

5.88

ⓐ $(\frac{f}{g}) (x) = x - 9$
ⓑ undefined

5.89

$−1$

5.90

$6$

5.91

yes

5.92

yes

Section 5.1 Exercises

1.

ⓐ trinomial, 5 ⓑ polynomial, 3 ⓒ binomial, 2 ⓓ monomial, 0
ⓔ binomial, 2

3.

ⓐ binomial, 1ⓑ trinomial, 2
ⓒ polynomial, 3ⓓ trinomial, 5
ⓔ monomial, 0

5.

ⓐ $trinomial, 2$ ⓑ $monomial, 4$ ⓒ $binomial, 1$ ⓓ $polynomial, 3$
ⓔ $monomial, 0$

7.

ⓐ $binomial, 1$ ⓑ $trinomial, 2$ ⓒ $polynomial, 3$ ⓓ $binomial, 3$
ⓔ $monomial, 0$

9.

ⓐ ${12x}^{2}$ ⓑ $− 5a$

11.

ⓐ $6 w$ ⓑ $19 x^{2} y$

13.

${12x}^{2} - 5a$

15.

$6 w + 19 x^{2} y$

17.

ⓐ $−22 b$ ⓑ $16 x y$

19.

ⓐ $−10 a + 5 b$
ⓑ $p q^{2} - 4 p - 3 q^{2}$

21.

ⓐ $−4 a^{2} + b^{2}$
ⓑ $x^{2} y - 3 x + 7 x y^{2}$

23.

ⓐ $x y^{2} - 5 x - 5 y^{2}$
ⓑ $19 y + 5 z$

25.

$x - 3 y + a^{2} b - 4 a - 5 a b^{2}$

27.

$− u^{2} + 4 v^{2} + 12a + 8 b$

29.

$− 4a - 3 b$

31.

$−17 x^{6}$

33.

$11 y^{2} + 4 y + 11$

35.

$−3 x^{2} + 17 x - 1$

37.

$11 x^{2} - 5 x + 5$

39.

$6 a^{2} - 4 a - 1$

41.

$2 m^{2} - 7 m + 4$

43.

$11 a + 3$

45.

$12 s^{2} - 16 s + 9$

47.

$3 x^{2} - x + 4$

49.

$w^{2} + 3 w + 4$

51.

$11 w - 66$

53.

$10 x^{2} - 7 x y + 6 y^{2}$

55.

$10 m^{2} + 3 m n - 8 n^{2}$

57.

$−3 a b + 3 b^{2}$

59.

$p^{3} - 6 p^{2} q + p q^{2} + 4 q^{3}$

61.

$x^{3} + 2 x^{2} y - 5 x y^{2} + y^{3}$

63.

ⓐ 187 ⓑ 40 ⓒ 2

65.

ⓐ $−104$ ⓑ 4 ⓒ 40

67.

The height is 11 feet.

69.

The revenue is $10,800.

71.

The cost is $456.

73.

ⓐ $(f + g) (x) = 7 x^{2} + 4 x + 4$ ⓑ $(f + g) (2) = 40$
ⓒ $(f - g) (x) = −3 x^{2} - 12 x - 2$
ⓓ $(f - g) (−3) = 7$

75.

ⓐ $(f + g) (x) = 6 x^{3} - x^{2} - 9 x + 3$
ⓑ $(f + g) (2) = 29$
ⓒ $(f - g) (x) = − x^{2} + 5 x + 3$
ⓓ $(f - g) (−3) = −21$

77.

Answers will vary.

79.

Answers will vary.

Section 5.2 Exercises

81.

ⓐ $d^{9}$ ⓑ $4^{14 x}$ ⓒ $8 y^{4}$ ⓓ $w^{6}$

83.

ⓐ $n^{31}$ ⓑ $3^{x + 6}$ ⓒ $56 w^{6}$
ⓓ $a^{16}$

85.

$m^{x + 3}$

87.

$y^{a + b}$

89.

ⓐ $x^{15}$ ⓑ $5^{9}$ ⓒ $\frac{1}{q^{18}}$ ⓓ $\frac{1}{10}$

91.

ⓐ $p^{14}$ ⓑ $4^{12}$ ⓒ $\frac{1}{b^{8}}$ ⓓ $\frac{1}{4^{5}}$

93.

ⓐ 1 ⓑ 1

95.

ⓐ $−1$ ⓑ $−1$

97.

ⓐ $\frac{1}{a^{2}}$ ⓑ $\frac{1}{1000}$ ⓒ $c^{5}$ ⓓ $9$

99.

ⓐ $\frac{1}{r^{3}}$ ⓑ $\frac{1}{100,000}$ ⓒ $q^{10}$
ⓓ $1,000$

101.

ⓐ $\frac{64}{25}$ ⓑ $\frac{a^{2}}{b^{2}}$

103.

ⓐ $\frac{729}{64}$ ⓑ $- \frac{v^{5}}{u^{5}}$

105.

ⓐ $\frac{1}{25}$ ⓑ $- \frac{1}{25}$ ⓒ $25$ ⓓ $−25$

107.

ⓐ $\frac{3}{5}$ ⓑ $\frac{1}{15}$

109.

ⓐ $\frac{1}{b^{4}}$ ⓑ $\frac{w^{2}}{x^{9}}$ ⓒ $−12 c d^{4}$

111.

ⓐ 1 ⓑ $\frac{1}{u^{4} v^{5}}$ ⓒ $\frac{- 36 r^{2}}{s^{5}}$

113.

$\frac{1}{p}$

115.

ⓐ $m^{8}$ ⓑ $10^{18}$ ⓒ $\frac{1}{x^{12}}$

117.

ⓐ $y^{3 x}$ ⓑ $5^{x y}$ ⓒ $\frac{1}{q^{48}}$

119.

ⓐ $9 x^{2} y^{2}$ ⓑ 1 ⓒ $\frac{1}{25 x^{4}}$
ⓓ $\frac{16}{y^{6}}$

121.

ⓐ $−125 a^{3} b^{3}$ ⓑ 1 ⓒ $\frac{1}{36 x^{6}}$ ⓓ $\frac{9}{y^{8}}$

123.

ⓐ $\frac{p^{5}}{32}$ ⓑ $\frac{y^{6}}{x^{6}}$ ⓒ $\frac{8 x^{3} y^{6}}{z^{3}}$
ⓓ $\frac{16}{p^{6} q^{4}}$

125.

ⓐ $\frac{a^{4}}{81 b^{4}}$ ⓑ $\frac{16 m^{2}}{25}$ ⓒ $\frac{a^{4} c^{4}}{9 b^{6}}$ ⓓ $\frac{q^{8} r^{8}}{p^{2}}$

127.

ⓐ $1125 t^{8}$ ⓑ $\frac{1}{t^{19}}$ ⓒ $\frac{y^{4}}{3 x^{2}}$

129.

ⓐ $16 m^{8} n^{22}$ ⓑ $\frac{4}{p^{6}}$

131.

ⓐ $\frac{7}{n}$ ⓑ $\frac{1}{7 n}$ ⓒ $- \frac{1}{7 n}$

133.

ⓐ $\frac{1}{9 p^{2}}$ ⓑ $\frac{3}{p^{2}}$ ⓒ $\frac{−3}{p^{2}}$

135.

$x^{14}$

137.

$a^{36}$

139.

$8 m^{18}$

141.

$1,000 x^{6} y^{3}$

143.

$16 a^{12} b^{8}$

145.

$\frac{8}{27} x^{6} y^{3}$

147.

$1,024 a^{10}$

149.

$25,000 p^{24}$

151.

$x^{18} y^{18}$

153.

$144 m^{8} n^{22}$

155.

ⓐ $45 x^{3}$ ⓑ $48 y^{4}$

157.

ⓐ $\frac{1}{2 r^{4}}$ ⓑ $\frac{1}{3} x^{11}$

159.

$\frac{1}{j^{3}}$

161.

$- \frac{4000}{n^{12}}$

163.

ⓐ $3.4 \times 10^{5}$ ⓑ $4.1 \times 10^{−2}$

165.

ⓐ $1.29 \times 10^{6}$
ⓑ $1.03 \times 10^{−6}$

167.

ⓐ $−830$ ⓑ 0.038

169.

ⓐ 16,000,000,000
ⓑ 0.00000843

171.

ⓐ 0.02 ⓑ 500,000,000

173.

ⓐ 0.0000056 ⓑ 20,000,000

175.

Answers will vary.

177.

Answers will vary.

Section 5.3 Exercises

179.

ⓐ $30 x^{8}$ ⓑ $15 x^{8} y^{2}$

181.

ⓐ $72 c^{5}$ ⓑ $\frac{1}{3} m^{5} n^{5}$

183.

ⓐ $−5 t^{3} - 15 t^{2} + 90 t$
ⓑ $9 s r^{5} - 27 s^{2} r^{4} + 45 s^{3} r^{3}$

185.

ⓐ $−5 m^{3} - 15 m^{2} + 90 m$
ⓑ $−21 x^{4} y^{2} - 30 x^{3} y^{3} + 3 x^{2} y^{4}$

187.

$y^{2} + 12 y + 27$

189.

$21 q^{2} - 44 q - 32$

191.

$y^{2} - 8 y + 12$

193.

$6 p^{2} + 11 p + 5$

195.

$m^{2} + 7 m - 44$

197.

$33 r^{2} - 85 r - 8$

199.

$y^{3} + 3 y^{2} - 4 y - 12$

201.

$6 x^{2} y^{2} + 13 x y + 6$

203.

$y^{4} - 11 y^{2} + 28$

205.

$9 r^{2} s^{2} - 33 r s + 28$

207.

$u^{3} + 7 u^{2} + 14 u + 8$

209.

$3 a^{3} + 31 a^{2} + 5 a - 50$

211.

$6 a^{4} - 13 a^{3} + 15 a^{2} + 35 a - 50$

213.

$p^{3} - 10 p^{2} + 33 p - 36$

215.

$6 r^{3} - 41 r^{2} - 61 r - 9$

217.

$q^{2} + 24 q + 144$

219.

$4 y^{2} - 12 y z + 9 z^{2}$

221.

$x^{2} + \frac{4}{3} x + \frac{4}{9}$

223.

$\frac{1}{64} x^{2} - \frac{1}{36} x y + \frac{1}{81} y^{2}$

225.

$25 u^{4} + 90 u^{2} + 81$

227.

$64 p^{6} - 48 p^{3} + 9$

229.

$64 j^{2} - 16$

231.

$81 c^{2} - 25$

233.

$49 w^{2} - 100 x^{2}$

235.

$p^{2} - \frac{16}{25} q^{2}$

237.

$x^{2} y^{2} - 81$

239.

$225 m^{4} - 64 n^{8}$

241.

$t^{2} - 18 t + 81$

243.

$2 x^{2} - 3 x y - 2 y^{2}$

245.

$9 p^{2} - 64$

247.

$k^{2} - 12 k + 36$

249.

$8 x^{3} - x^{2} y^{2} + 64 x y - 8 y^{3}$

251.

$y^{8} + 4 y^{4} z + 4 z^{2}$

253.

$m^{6} - 16 m^{3} n + 64 n^{2}$

255.

$r^{5} + r^{2} s^{2} - r^{3} s^{3} - s^{5}$

257.

$18 p - 9$

259.

$−10 j - 15$

261.

$9 d^{7}$

263.

$42 m^{5} n^{8}$

265.

$5 q^{5} - 10 q^{4} + 30 q^{3}$

267.

$y^{3} - y^{2} - 2 y$

269.

$6 k^{3} + 11 k^{2} - 26 k + 4$

271.

$121 - b^{2}$

273.

$4 x^{4} - 9 y^{8}$

275.

$9 d^{2} + 6 d + 1$

277.

$9 z^{2} + \frac{6}{5} z + \frac{1}{25}$

279.

ⓐ $(f \cdot g) (x) = 4 x^{3} - x^{2} - 8 x + 5$
ⓑ $(f \cdot g) (−2) = −15$

281.

ⓐ $(f \cdot g) (x) = 49 x^{2} - 64$
ⓑ $(f \cdot g) (−2) = 132$

283.

ⓐ $(f \cdot g) (x) = x^{4} + 6 x^{3} + 9 x^{2} + 10 x - 12$ ⓑ $(f \cdot g) (1) = 14$

285.

Answers will vary.

287.

Answers will vary.

Section 5.4 Exercises

289.

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

291.

$\frac{−3 y^{3}}{4 x^{3}}$

293.

$\frac{−3 q^{5}}{p^{5}}$

295.

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

297.

$4 x^{2} + 3 x$

299.

$−6 y^{2} + 3 y$

301.

$6 r^{3} + 11 r^{2} s - 8 r s^{3}$

303.

$−5 y - 3 + \frac{1}{4 y}$

305.

$a - 7$

307.

$4 x + 3$

309.

$p + 3 - \frac{8}{p + 8}$

311.

$2 n^{2} - 6 n + 8 + \frac{4}{n + 3}$

313.

$m^{2} - 10 m + 100$

315.

$25 y^{2} + 20 y + 16$

317.

$x^{2} - 5 x + 6; 0$

319.

$2 x^{2} - 3 x + 4; 4$

321.

$x^{3} - 2 x^{2} + 5 x - 4; − 2$

323.

$3 x^{3} - 2 x^{2} - 4 x - 2; 0$

325.

ⓐ $(\frac{f}{g}) (x) = x - 6$
ⓑ $(\frac{f}{g}) (−5) = −11$

327.

ⓐ $(\frac{f}{g}) (x) = x^{2} + 5 x - 4$
ⓑ $(\frac{f}{g}) (0) = −4$

329.

ⓐ $(\frac{f}{g}) (x) = x - 1; x \neq - 3$ ⓑ $(\frac{f}{g}) (3) = 2$

331.

$−9$

333.

−6

335.

no

337.

yes

339.

answer will vary

341.

Answers will vary.

Review Exercises

343.

binomial

345.

trinomial

347.

$−13 y^{3}$

349.

$6 m^{2} + 19 m - 4$

351.

$5 u^{2} + 4 u + 7$

353.

$14 x - 7$

355.

$−7 b - 3 a$

357.

$8 a^{2} - 5 a - 2$

359.

$3 m^{2} - 3 m + 3$

361.

$3 y^{2} - 8 y - 14$

363.

$x^{3} + 2 x^{2} y - 5 x y^{2} + y^{3}$

365.

ⓐ 165 ⓑ 39 ⓒ 5

367.

The height is $64 feet$ .

369.

371.

$p^{13}$

373.

$a^{6}$

375.

$y^{a + b}$

377.

$a^{5}$

379.

$\frac{1}{x^{5}}$

381.

1

383.

$−3$

385.

$- \frac{1}{1000}$

387.

$\frac{1}{8 n}$

389.

$\frac{1}{1000}$

391.

36

393.

$−125$

395.

$- \frac{1}{125}$

397.

$\frac{{- x}^{3}}{27}$

399.

$3^{10}$

401.

$x^{6}$

403.

$\frac{1}{u^{3} v^{5}}$

405.

$\frac{1}{p}$

407.

$\frac{1}{q^{16}}$

409.

$n^{2}$

411.

1

413.

$\frac{9}{y^{8}}$

415.

$\frac{81 x^{4} y^{8}}{z^{4}}$

417.

$27 x^{7} y^{17}$

419.

$\frac{3 y^{4}}{4 x^{4}}$

421.

$5.3 \times 10^{6}$

423.

$29,000$

425.

$0.00009413$

427.

$0.00072$

429.

$9,000$

431.

$10 c^{10}$

433.

$\frac{m^{7} n^{10}}{9}$

435.

$a^{4} - 9 a^{3} - 36 a^{2}$

437.

$8 n^{4} - 10 n^{3}$

439.

$y^{2} + 8 y - 48$

441.

$18 p^{2} - 93 p + 110$

443.

$k^{2} - 3 k - 54$

445.

$10 y^{2} - 59 y + 63$

447.

$x^{2} + x - 72$

449.

$30 a^{2} - 33 a + 3$

451.

$15 b^{3} - b^{2} - 47 b + 18$

453.

$24 y^{3} - 54 y^{2} + 32 y - 5$

455.

$x^{2} + \frac{3}{2} x + \frac{9}{16}$

457.

$25 p^{2} + 70 p q + 49 q^{2}$

459.

$36 x^{2} - y^{2}$

461.

$144 x^{6} - 49 y^{4}$

463.

$9 p^{9}$

465.

$- \frac{3}{y^{4}}$

467.

$\frac{4 a^{5}}{b^{2}}$

469.

$\frac{4 m^{9}}{n^{4}}$

471.

$−9 y^{2} + 4 y$

473.

$−3 x - 1 + \frac{3}{4 x}$

475.

$y - 3 + \frac{33}{y + 5}$

477.

$a^{2} - a + 1 - \frac{2}{a + 1}$

479.

$2 x^{2} - 5 x - 4; 0$

481.

ⓐ $(\frac{f}{g}) (x) = x - 6$
ⓑ $(\frac{f}{g}) (−2) = −8$

483.

$−9$

485.

no

Practice Test

487.

ⓐ trinomial ⓑ 4

489.

$6 x^{2} - 3 x + 11$

491.

$x$

493.

$1$

495.

$\frac{1}{8 y^{3}}$

497.

$x^{9}$

499.

$\frac{4 r}{s^{6}}$

501.

$−48 x^{5} y^{9}$

503.

$7 m^{2} + 19 m - 6$

505.

$16 x^{2} - 24 x + 9$

507.

$3 y^{2} - 7 x$

509.

yes

511.

$4.8 \times 10^{3}$

513.

515.

ⓐ $(\frac{f}{g}) (x) = 3 x + 4$
ⓑ $(\frac{f}{g}) (3) = 13$