Chapter 6

Be Prepared

6.1

$2 \cdot 2 \cdot 2 \cdot 7$

6.2

6.3

$−21 a^{2} - 24 a b$

6.4

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

6.5

$6 y^{2} + 23 y + 20$

6.6

−54, 54

6.7

$27 x^{6}$

6.8

$m^{2} + 8 m + 16$

6.9

$x^{2} - 9$

6.10

$y = \frac{3}{5}$

6.11

$n (n - 11) (n + 2)$

6.12

$8; x = 2$

Try It

6.1

$5 m^{2}$

6.2

$7 x$

6.3

$3 y^{2} (3 x + 2 x^{2} + 7 y)$

6.4

$3 p (p^{2} - 2 p q + 3 q^{2})$

6.5

$2 x^{2} (x + 6)$

6.6

$3 y^{2} (2 y - 5)$

6.7

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

6.8

$2 a b (4 a^{2} + a b - 3 b^{2})$

6.9

$−4 b (b^{2} - 4 b + 2)$

6.10

$−7 a (a^{2} - 3 a + 2)$

6.11

$(m + 3) (4 m - 7)$

6.12

$(n - 4) (8 n + 5)$

6.13

$(x + 8) (y + 3)$

6.14

$(a + 7) (b + 8)$

6.15

ⓐ $(x - 5) (x + 2)$
ⓑ $(5 x - 4) (4 x - 3)$

6.16

ⓐ $(y + 4) (y - 7)$
ⓑ $(7 m - 3) (6 m - 5)$

6.17

$(q + 4) (q + 6)$

6.18

$(t + 2) (t + 12)$

6.19

$(u - 3) (u - 6)$

6.20

$(y - 7) (y - 9)$

6.21

$(m + 3) (m + 6)$

6.22

$(n - 3) (n - 4)$

6.23

$(a - b) (a - 10 b)$

6.24

$(m - n) (m - 12 n)$

6.25

prime

6.26

prime

6.27

$5 x (x - 1) (x + 4)$

6.28

$6 y (y - 2) (y + 5)$

6.29

$(a + 1) (2 a + 3)$

6.30

$(b + 1) (4 b + 1)$

6.31

$(2 x - 3) (4 x - 1)$

6.32

$(2 y - 7) (5 y - 1)$

6.33

$(3 x + 2 y) (6 x - 5 y)$

6.34

$(3 x + y) (10 x - 21 y)$

6.35

$5 n (n - 4) (3 n - 5)$

6.36

$8 q (q + 6) (7 q - 2)$

6.37

$(x + 2) (6 x + 1)$

6.38

$(2 y + 1) (2 y + 3)$

6.39

$4 (2 x - 3) (2 x - 1)$

6.40

$3 (3 w - 2) (2 w - 3)$

6.41

$(h^{2} - 2) (h^{2} + 6)$

6.42

$(y^{2} + 4) (y^{2} - 5)$

6.43

$(x - 3) (x - 1)$

6.44

$(y - 1) (y + 1)$

6.45

${(2 x + 3)}^{2}$

6.46

${(3 y + 4)}^{2}$

6.47

${(8 y - 5)}^{2}$

6.48

${(4 z - 9)}^{2}$

6.49

${(7 x + 6 y)}^{2}$

6.50

${(8 m + 7 n)}^{2}$

6.51

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

6.52

$3 q {(3 p + 5)}^{2}$

6.53

$(11 m - 1) (11 m + 1)$

6.54

$(9 y - 1) (9 y + 1)$

6.55

$(14 m - 5 n) (14 m + 5 n)$

6.56

$(11 p - 3 q) (11 p + 3 q)$

6.57

$2 y^{2} (x - 2) (x + 2) (x^{2} + 4)$

6.58

$7 c^{2} (a - b) (a + b) (a^{2} + b^{2})$

6.59

$(x - 5 - y) (x - 5 + y)$

6.60

$(x + 3 - 2 y) (x + 3 + 2 y)$

6.61

$(x + 3) (x^{2} - 3 x + 9)$

6.62

$(y + 2) (y^{2} - 2 y + 4)$

6.63

$(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})$

6.64

$125 (4 m^{2} + 2 m n + n^{2}) (2 m - n)$

6.65

$4 (5 p + q) (25 p^{2} - 5 p q + q^{2})$

6.66

$2 (6 c + 7 d) (36 c^{2} - 42 c d + 49 d^{2})$

6.67

$(−2 y + 1) (13 y^{2} + 5 y + 1)$

6.68

$(−4 n + 3) (31 n^{2} + 21 n + 9)$

6.69

$8 y (y - 1) (y + 3)$

6.70

$5 y (y - 9) (y + 6)$

6.71

$4 x (2 x - 3) (2 x + 3)$

6.72

$3 (3 y - 4) (3 y + 4)$

6.73

${(2 x + 5 y)}^{2}$

6.74

${(3 x - 4 y)}^{2}$

6.75

$2 x y (25 x^{2} + 36)$

6.76

$3 x y (9 y^{2} + 16)$

6.77

$2 (5 m + 6 n) (25 m^{2} - 30 m n + 36 n^{2})$

6.78

$2 (p + 3 q) (p^{2} - 3 p q + 9 q^{2})$

6.79

$4 a b (a^{2} + 4) (a - 2) (a + 2)$

6.80

$7 x y (y^{2} + 1) (y - 1) (y + 1)$

6.81

$6 (x + b) (x - 2 c)$

6.82

$2 (4 x - 1) (2 x + 3 y)$

6.83

$4 q (p - 3) (p - 1)$

6.84

$3 p (2 q + 1) (q - 2)$

6.85

$(2 x - 3 y - 5) (2 x - 3 y + 5)$

6.86

$(4 x - 3 y - 8) (4 x - 3 y + 8)$

6.87

$m = \frac{2}{3}, m = - \frac{1}{2}$

6.88

$p = - \frac{3}{4}, p = \frac{3}{4}$

6.89

$c = 2, c = \frac{4}{3}$

6.90

$d = 3, d = - \frac{1}{2}$

6.91

$p = \frac{7}{5}, p = - \frac{7}{5}$

6.92

$x = \frac{11}{6}, x = - \frac{11}{6}$

6.93

$m = 1, m = \frac{3}{2}$

6.94

$k = 3, k = −3$

6.95

$a = - \frac{5}{2}, a = \frac{2}{3}$

6.96

$b = −2, b = - \frac{1}{20}$

6.97

$x = 0, x = \frac{3}{2}$

6.98

$y = 0, y = \frac{1}{4}$

6.99

ⓐ $x = −3$ or $x = 5$
ⓑ $(−3, 7)$ $(5, 7)$

6.100

ⓐ $x = 1$ or $x = 7$
ⓑ $(1, −4)$ $(7, −4)$

6.101

ⓐ $x = 1$ or $x = \frac{5}{2}$
ⓑ $(1, 0),$ $(\frac{5}{2}, 0)$ ⓒ $(0, 5)$

6.102

ⓐ $x = −3$ or $x = \frac{5}{6}$
ⓑ $(−3, 0),$ $(\frac{5}{6}, 0)$ ⓒ $(0, −15)$

6.103

$−15, −17$ and 15, 17

6.104

$−23, −21$ and 21, 23

6.105

The width is 5 feet and length is 6 feet.

6.106

The width of the patio is 12 feet and the length is 15 feet.

6.107

5 feet and 12 feet

6.108

The other leg is 24 feet and the hypotenuse is 25 feet.

6.109

ⓐ 5 seconds; ⓑ 0 and 3 seconds; ⓒ 196 feet

6.110

ⓐ 4 seconds; ⓑ 0 and 2 seconds; ⓒ 144 feet

Section 6.1 Exercises

$2 p q$

$6 m^{2} n^{3}$

$2 a$

$5 x^{3} y$

$3 (2 m + 3)$

11.

$9 (n - 7)$

13.

$3 (x^{2} + 2 x - 3)$

15.

$2 (4 p^{2} + 2 p + 1)$

17.

$8 y^{2} (y + 2)$

19.

$5 x (x^{2} - 3 x + 4)$

21.

$3 x (8 x^{2} - 4 x + 5)$

23.

$6 y^{2} (2 x + 3 x^{2} - 5 y)$

25.

$4 x y (5 x^{2} - x y + 3 y^{2})$

27.

$−2 (x + 2)$

29.

$−2 x (x^{2} - 9 x + 4)$

31.

$−4 p q (p^{2} + 3 p q - 4 q)$

33.

$(x + 1) (5 x + 3)$

35.

$(b - 2) (3 b - 13)$

37.

$(b + 5) (a + 3)$

39.

$(y + 5) (8 y + 1)$

41.

$(u + 2) (v - 9)$

43.

$(u - 1) (u + 6)$

45.

$(3 p - 5) (3 p + 4)$

47.

$(n - 6) (m - 4)$

49.

$(x - 7) (2 x - 5)$

51.

$−9 x y (2 y + 3 x)$

53.

$(x^{2} + 2) (3 x - 7)$

55.

$(x + y) (x + 5)$

57.

Answers will vary.

59.

Answers will vary.

Section 6.2 Exercises

61.

$(p + 5) (p + 6)$

63.

$(n + 3) (n + 16)$

65.

$(a + 5) (a + 20)$

67.

$(x - 2) (x - 6)$

69.

$(y - 3) (y - 15)$

71.

$(x - 1) (x - 7)$

73.

$(p - 1) (p + 6)$

75.

$(x - 4) (x - 2)$

77.

$(x - 12) (x + 1)$

79.

$(x + 8 y) (x - 10 y)$

81.

$(m + n) (m - 65 n)$

83.

$(a + 8 b) (a - 3 b)$

85.

Prime

87.

Prime

89.

$p (p - 10) (p + 2)$

91.

$3 m (m - 5) (m - 2)$

93.

$5 x^{2} (x - 3) (x + 5)$

95.

$(2 t + 5) (t + 1)$

97.

$(11 x + 1) (x + 3)$

99.

$(4 w - 1) (w - 1)$

101.

$(4 q + 1) (q - 2)$

103.

$(2 p - 5 q) (3 p - 2 q)$

105.

$(4 a - 3 b) (a + 5 b)$

107.

$−16 (x + 1) (x + 1)$

109.

$−10 q (3 q + 2) (q + 4)$

111.

$(5 n + 1) (n + 4)$

113.

$(2 k - 3) (2 k - 5)$

115.

$(3 y + 5) (2 y - 3)$

117.

$(2 n + 3) (n - 15)$

119.

$10 (6 y - 1) (y + 5)$

121.

$3 z (8 z + 3) (2 z - 5)$

123.

$8 (2 s + 3) (s + 1)$

125.

$12 (4 y - 3) (y + 1)$

127.

$(x^{2} + 1) (x^{2} - 7)$

129.

$(x^{2} - 7) (x^{2} + 4)$

131.

$(x - 12) (x + 1)$

133.

$(3 y - 4) (3 y - 1)$

135.

$(u - 6) (u - 6)$

137.

$(r - 4 s) (r - 16 s)$

139.

$(4 y - 7) (3 y - 2)$

141.

$(2 n - 1) (3 n + 4)$

143.

$13 (z^{2} + 3 z - 2)$

145.

$3 p (p + 7)$

147.

$6 (r + 2) (r + 3)$

149.

$4 (2 n + 1) (3 n + 1)$

151.

$(x^{2} + 2) (x^{2} - 6)$

153.

$(x - 9) (x + 6)$

155.

Answers will vary.

157.

Answers will vary.

Section 6.3 Exercises

159.

${(4 y + 3)}^{2}$

161.

${(6 s + 7)}^{2}$

163.

${(10 x - 1)}^{2}$

165.

${(5 n - 12)}^{2}$

167.

${(7 x + 2 y)}^{2}$

169.

$(10 y - 1)^{2}$

171.

$10 j {(k + 4)}^{2}$

173.

$3 u^{2} {(5 u - v)}^{2}$

175.

$(5 v - 1) (5 v + 1)$

177.

$(2 - 7 x) (2 + 7 x)$

179.

$6 p^{2} (q - 3) (q + 3)$

181.

$6 (4 p^{2} + 9)$

183.

$(11 x - 12 y) (11 x + 12 y)$

185.

$(13 c - 6 d) (13 c + 6 d)$

187.

$(2 z - 1) (2 z + 1) (4 z^{2} + 1)$

189.

$2 b^{2} (3 a - 2) (3 a + 2) (9 a^{2} + 4)$

191.

$(x - 8 - y) (x - 8 + y)$

193.

$(a + 3 - 3 b) (a + 3 + 3 b)$

195.

$(x + 5) (x^{2} - 5 x + 25)$

197.

$(z^{2} - 3) (z^{4} + 3 z^{2} + 9)$

199.

$(2 - 7 t) (4 + 14 t + 49 t^{2})$

201.

$(2 y - 5 z) (4 y^{2} + 10 y z + 25 z^{2})$

203.

$(6 a + 5 b) (36 a^{2} - 30 a b + 25 b^{2})$

205.

$7 (k + 2) (k^{2} - 2 k + 4)$

207.

$2 x^{2} (1 - 2 y) (1 + 2 y + 4 y^{2})$

209.

$9 (x + 1) (x^{2} + 3)$

211.

$− (3 y + 5) (21 y^{2} - 30 y + 25)$

213.

$(8 a - 5) (8 a + 5)$

215.

$3 (3 q - 1) (3 q + 1)$

217.

${(4 x - 9)}^{2}$

219.

$2 (4 p^{2} + 1)$

221.

$(5 - 2 y) (25 + 10 y + 4 y^{2})$

223.

$5 {(3 n + 2)}^{2}$

225.

$(x - 5 - y) (x - 5 + y)$

227.

$(3 x + 1) (3 x^{2} + 1)$

229.

Answers will vary.

231.

Answers will vary.

Section 6.4 Exercises

233.

$(2 n - 1) (n + 7)$

235.

$a^{3} (a^{2} + 9)$

237.

$(11 r - s) (11 r + s)$

239.

$8 (m - 2) (m + 2)$

241.

${(5 w - 6)}^{2}$

243.

${(m + 7 n)}^{2}$

245.

$7 (b + 3) (b - 2)$

247.

$3 x y (x - 3) (x^{2} + 3 x + 9)$

249.

$(k - 2) (k + 2) (k^{2} + 4)$

251.

$5 x y^{2} (x^{2} + 4) (x + 2) (x - 2)$

253.

$3 (5 p + 4) (q - 1)$

255.

$4 (x + 3) (x + 7)$

257.

$4 u^{2} (u + v) (u^{2} - u v + v^{2})$

259.

prime

261.

$10 (m - 5) (m + 5) (m^{2} + 25)$

263.

$3 y (3 x + 2) (4 x - 1)$

265.

$(2 x - 3 y) (4 x^{2} + 6 x y + 9 y^{2})$

267.

$(y + 1) (y - 1) (y^{2} - y + 1) (y^{2} + y + 1)$

269.

$(3 x - y + 7) (3 x - y - 7)$

271.

$(3 x - 2)^{2}$

273.

Answers will vary.

275.

Answers will vary.

Section 6.5 Exercises

277.

$a = 10 / 3, a = 7 / 2$

279.

$m = 0, m = 5 / 12$

281.

$x = 1 / 2$

283.

$a = \frac{−4}{5}, a = 6$

285.

$m = 5 / 4, m = 3$

287.

$a = −1, a = 0$

289.

$m = 12 / 7, m = −12 / 7$

291.

$y = −9 / 4, y = 9 / 4$

293.

$n = −6 / 11, n = 6 / 11$

295.

$x = 2, x = −5$

297.

$x = 3 / 2, x = −1$

299.

$x = 2, x = −4 / 3$

301.

$x = 3 / 2$

303.

$x = 2, x = −4 / 3$

305.

$x = −3 / 2, x = 1 / 3$

307.

$p = 0, p = ¾$

309.

$x = 0, x = 6$

311.

$x = 0, x = –1 / 3$

313.

ⓐ $x = 2$ or $x = 6$ ⓑ $(2, −4)$ $(6, −4)$

315.

ⓐ $x = \frac{3}{2}$ or $x = \frac{3}{4}$
ⓑ $(\frac{3}{2}, −4)$ $(\frac{3}{4}, −4)$

317.

319.

321.

$−13, −11$ and 11, 13

323.

$−14, −12$ and 12, 14

325.

Width: 4 feet; Length: 7 feet.

327.

Width: 5 feet; Length: 11 feet.

329.

The sides are 6 feet and 8 feet.

331.

The building side is 8 feet, the hypotenuse is 17 feet, and the third side is 15 feet.

333.

ⓐ 0 seconds and 2 seconds ⓑ 1 second

335.

Answers will vary.

Review Exercises

337.

$3 a b^{2}$

339.

$3 y$

341.

$7 (5 y + 12)$

343.

$3 x (6 x^{2} - 5)$

345.

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

347.

$−3 x (x^{2} - 9 x + 4)$

349.

$(a + b) (x - y)$

351.

$(x - 3) (x + 7)$

353.

$(m^{2} + 1) (m + 1)$

355.

$(a + 3) (a + 11)$

357.

$(m + 9) (m - 6)$

359.

$(x + 5 y) (x + 7 y)$

361.

$(a + 7 b) (a - 3 b)$

363.

Prime

365.

$3 y (y - 5) (y - 2)$

367.

$(5 y + 9) (y + 1)$

369.

$(5 y + 1) (2 y - 11)$

371.

$−9 (9 a + 1) (a - 2)$

373.

$(3 a - 1) (6 a - 1)$

375.

Prime

377.

$3 (x + 4) (x - 3)$

379.

$3 (2 a - 7) (3 a + 1)$

381.

$(x^{2} - 15) (x^{2} + 2)$

383.

${(5 x + 3)}^{2}$

385.

$10 {(2 x + 9)}^{2}$

387.

$3 u^{2} {(5 u - v)}^{2}$

389.

$(13 m + n) (13 m - n)$

391.

$(3 + 11 y) (3 - 11 y)$

393.

$n (13 n + 1) (13 n - 1)$

395.

$6 (4 p^{2} + 9)$

397.

$(2 z - 1) (2 z + 1) (4 z^{2} + 1)$

399.

$(a + 3 - 3 b) (a + 3 + 3 b)$

401.

$(a - 5) (a^{2} + 5 a + 25)$

403.

$2 (m + 3) (m^{2} - 3 m + 9)$

405.

$4 x^{2} (6 x + 11)$

407.

${(4 n - 7 m)}^{2}$

409.

$5 u^{2} (u + 3) (u - 3)$

411.

prime

413.

$(b - 4) (b^{2} + 4 b + 16)$

415.

$(2 b + 5 c) (b - c)$

417.

$5 (q + 3) (q - 6)$

419.

$10 (m - 5) (m + 5) (m^{2} + 25)$

421.

$(4 x - 3 y + 8) (4 x - 3 y - 8)$

423.

$b = −1 / 5, b = −1 / 6$

425.

$x = 1 / 2$

427.

$x = −4, x = −5$

429.

$p = - \frac{5}{2}, p = 8$

431.

$m = \frac{5}{12}, m = - \frac{5}{12}$

433.

$x = 2, x = −5$

435.

$p = 0, p = ¾$

437.

ⓐ $x = −7$ or $x = −4$
ⓑ $(−7, −8)$ $(−4, −8)$

439.

441.

The numbers are $−21$ and $−19$ or 19 and 21.

443.

The lengths are 8, 15, and 17 ft.

Practice Test

445.

$40 a^{2} (2 + 3 a)$

447.

$(x + 4) (x + 9)$

449.

$(x - 8) (y + 7)$

451.

${(3 s - 2)}^{2}$

453.

$3 (x + 5 y) (x - 5 y)$

455.

$(x + 5) (x^{2} - 5 x + 25)$

457.

$(3 x^{2} - 5) (2 x^{2} - 3)$

459.

$a = 4 / 5, a = −6$

461.

The width is 12 inches and the length is 14 inches.

463.

ⓐ $x = 3$ or $x = 4$ ⓑ $(3, −7)$ $(4, −7)$