13.1.6: Chapter 6

Be Prepared

6.1

$2·2·2·7$

6.2

72

6.3

$\mathrm{-21}{a}^2-24ab$

6.4

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

6.5

$6y^2+23y+20$

6.6

−54, 54

6.7

$27x^6$

6.8

$m^2+8m+16$

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

$5m^2$

6.2

$7x$

6.3

$3y^2\left(3x+2x^2+7y\right)$

6.4

$3p\left(p^2-2pq+3q^2\right)$

6.5

$2x^2\left(x+6\right)$

6.6

$3y^2\left(2y-5\right)$

6.7

$3xy\left(5x^2-xy+2y^2\right)$

6.8

$2ab\left(4a^2+ab-3b^2\right)$

6.9

$\mathrm{-4}b\left(b^2-4b+2\right)$

6.10

$\mathrm{-7}a\left(a^2-3a+2\right)$

6.11

$\left(m+3\right)\left(4m-7\right)$

6.12

$\left(n-4\right)\left(8n+5\right)$

6.13

$\left(x+8\right)\left(y+3\right)$

6.14

$\left(a+7\right)\left(b+8\right)$

6.15

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

6.16

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

6.17

$\left(q+4\right)\left(q+6\right)$

6.18

$\left(t+2\right)\left(t+12\right)$

6.19

$\left(u-3\right)\left(u-6\right)$

6.20

$\left(y-7\right)\left(y-9\right)$

6.21

$\left(m+3\right)\left(m+6\right)$

6.22

$\left(n-3\right)\left(n-4\right)$

6.23

$\left(a-b\right)\left(a-10b\right)$

6.24

$\left(m-n\right)\left(m-12n\right)$

6.25

prime

6.26

prime

6.27

$5x\left(x-1\right)\left(x+4\right)$

6.28

$6y\left(y-2\right)\left(y+5\right)$

6.29

$\left(a+1\right)\left(2a+3\right)$

6.30

$\left(b+1\right)\left(4b+1\right)$

6.31

$\left(2x-3\right)\left(4x-1\right)$

6.32

$\left(2y-7\right)\left(5y-1\right)$

6.33

$\left(3x+2y\right)\left(6x-5y\right)$

6.34

$\left(3x+y\right)\left(10x-21y\right)$

6.35

$5n\left(n-4\right)\left(3n-5\right)$

6.36

$8q\left(q+6\right)\left(7q-2\right)$

6.37

$\left(x+2\right)\left(6x+1\right)$

6.38

$\left(2y+1\right)\left(2y+3\right)$

6.39

$4\left(2x-3\right)\left(2x-1\right)$

6.40

$3\left(3w-2\right)\left(2w-3\right)$

6.41

$\left(h^2-2\right)\left(h^2+6\right)$

6.42

$\left(y^2+4\right)\left(y^2-5\right)$

6.43

$\left(x-3\right)\left(x-1\right)$

6.44

$\left(y-1\right)\left(y+1\right)$

6.45

${\left(2x+3\right)}^2$

6.46

${\left(3y+4\right)}^2$

6.47

${\left(8y-5\right)}^2$

6.48

${\left(4z-9\right)}^2$

6.49

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

6.50

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

6.51

$2y{\left(2x-3\right)}^2$

6.52

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

6.53

$\left(11m-1\right)\left(11m+1\right)$

6.54

$\left(9y-1\right)\left(9y+1\right)$

6.55

$\left(14m-5n\right)\left(14m+5n\right)$

6.56

$\left(11p-3q\right)\left(11p+3q\right)$

6.57

$2y^2\left(x-2\right)\left(x+2\right)\left(x^2+4\right)$

6.58

$7c^2\left(a-b\right)\left(a+b\right)\left(a^2+b^2\right)$

6.59

$\left(x-5-y\right)\left(x-5+y\right)$

6.60

$\left(x+3-2y\right)\left(x+3+2y\right)$

6.61

$\left(x+3\right)\left(x^2-3x+9\right)$

6.62

$\left(y+2\right)\left(y^2-2y+4\right)$

6.63

$\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$

6.64

$125(2m-n)(4m^2+2mn+n^2)$

6.65

$4(5p+q)\left(25p^2-5pq+q^2\right)$

6.66

$2\left(6c+7d\right)\left(36c^2-42cd+49d^2\right)$

6.67

$\left(\mathrm{-2}y+1\right)\left(13y^2+5y+1\right)$

6.68

$\left(\mathrm{-4}n+3\right)\left(31n^2+21n+9\right)$

6.69

$8y\left(y-1\right)\left(y+3\right)$

6.70

$5y\left(y-9\right)\left(y+6\right)$

6.71

$4x\left(2x-3\right)\left(2x+3\right)$

6.72

$3\left(3y-4\right)\left(3y+4\right)$

6.73

${\left(2x+5y\right)}^2$

6.74

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

6.75

$2xy\left(25x^2+36\right)$

6.76

$3xy\left(9y^2+16\right)$

6.77

$2\left(5m+6n\right)\left(25m^2-30mn+36n^2\right)$

6.78

$2\left(p+3q\right)\left(p^2-3pq+9q^2\right)$

6.79

$4ab\left(a^2+4\right)\left(a-2\right)\left(a+2\right)$

6.80

$7xy\left(y^2+1\right)\left(y-1\right)\left(y+1\right)$

6.81

$6\left(x+b\right)\left(x-2c\right)$

6.82

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

6.83

$4q\left(p-3\right)\left(p-1\right)$

6.84

$3p\left(2q+1\right)\left(q-2\right)$

6.85

$\left(2x-3y-5\right)\left(2x-3y+5\right)$

6.86

$\left(4x-3y-8\right)\left(4x-3y+8\right)$

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

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

6.92

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

6.93

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

6.94

$k=3,k=\mathrm{-3}$

6.95

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

6.96

$b=\mathrm{-2},b=-\frac{1}{20}$

6.97

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

6.98

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

6.99

ⓐ $x=\mathrm{-3}$ or $x=5$
ⓑ $\left(\mathrm{-3},7\right)$ $\left(5,7\right)$

6.100

ⓐ $x=1$ or $x=7$
ⓑ $\left(1,\mathrm{-4}\right)$ $\left(7,\mathrm{-4}\right)$

6.101

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

6.102

ⓐ $x=\mathrm{-3}$ or $x=\frac{5}{6}$
ⓑ $\left(\mathrm{-3},0\right),$ $\left(\frac{5}{6},0\right)$ ⓒ $\left(0,\mathrm{-15}\right)$

6.103

$\mathrm{-15},\mathrm{-17}$ and 15, 17

6.104

$\mathrm{-23},\mathrm{-21}$ 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

1.

$2pq$

3.

$6m^2{n}^3$

5.

$2a$

7.

$5x^{3}y$

9.

$3\left(2m+3\right)$

11.

$9\left(n-7\right)$

13.

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

15.

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

17.

$8y^2(y+2)$

19.

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

21.

$3x\left(8x^2-4x+5\right)$

23.

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

25.

$4xy\left(5x^2-xy+3y^2\right)$

27.

$\mathrm{-2}\left(x+2\right)$

29.

$\mathrm{-2}x\left(x^2-9x+4\right)$

31.

$\mathrm{-4}pq\left(p^2+3pq-4q\right)$

33.

$\left(x+1\right)\left(5x+3\right)$

35.

$(b-2)\left(3b-13\right)$

37.

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

39.

$\left(y+5\right)\left(8y+1\right)$

41.

$\left(u+2\right)\left(v-9\right)$

43.

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

45.

$\left(3p-5\right)\left(3p+4\right)$

47.

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

49.

$\left(x-7\right)\left(2x-5\right)$

51.

$\mathrm{-9}xy\left(2y+3x\right)$

53.

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

55.

$\left(x+y\right)\left(x+5\right)$

57.

Answers will vary.

59.

Answers will vary.

Section 6.2 Exercises

61.

$\left(p+5\right)\left(p+6\right)$

63.

$\left(n+3\right)\left(n+16\right)$

65.

$\left(a+5\right)\left(a+20\right)$

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.

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

81.

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

83.

$\left(a+8b\right)\left(a-3b\right)$

85.

Prime

87.

Prime

89.

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

91.

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

93.

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

95.

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

97.

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

99.

$\left(4w-1\right)\left(w-1\right)$

101.

$\left(4q+1\right)\left(q-2\right)$

103.

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

105.

$\left(4a-3b\right)\left(a+5b\right)$

107.

$\mathrm{-16}\left(x+1\right)\left(x+1\right)$

109.

$\mathrm{-10}q\left(3q+2\right)\left(q+4\right)$

111.

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

113.

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

115.

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

117.

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

119.

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

121.

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

123.

$8\left(2s+3\right)\left(s+1\right)$

125.

$12\left(4y-3\right)\left(y+1\right)$

127.

$\left(x^2+1\right)\left(x^2-7\right)$

129.

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

131.

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

133.

$\left(3y-4\right)\left(3y-1\right)$

135.

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

137.

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

139.

$\left(4y-7\right)\left(3y-2\right)$

141.

$\left(2n-1\right)\left(3n+4\right)$

143.

$13\left(z^2+3z-2\right)$

145.

$3p\left(p+7\right)$

147.

$6\left(r+2\right)\left(r+3\right)$

149.

$4\left(2n+1\right)\left(3n+1\right)$

151.

$\left(x^2+2\right)\left(x^2-6\right)$

153.

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

155.

Answers will vary.

157.

Answers will vary.

Section 6.3 Exercises

159.

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

161.

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

163.

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

165.

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

167.

${\left(7x+2y\right)}^2$

169.

${\left(10y-1\right)}^2$

171.

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

173.

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

175.

$\left(5v-1\right)\left(5v+1\right)$

177.

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

179.

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

181.

$6\left(4p^2+9\right)$

183.

$\left(11x-12y\right)\left(11x+12y\right)$

185.

$\left(13c-6d\right)\left(13c+6d\right)$

187.

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

189.

$2b^2\left(3a-2\right)\left(3a+2\right)\left(9a^2+4\right)$

191.

$\left(x-8-y\right)\left(x-8+y\right)$

193.

$\left(a+3-3b\right)\left(a+3+3b\right)$

195.

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

197.

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

199.

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

201.

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

203.

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

205.

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

207.

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

209.

$9\left(x+1\right)\left(x^2+3\right)$

211.

$\text{−}\left(3y+5\right)\left(21y^2-30y+25\right)$

213.

$\left(8a-5\right)\left(8a+5\right)$

215.

$3\left(3q-1\right)\left(3q+1\right)$

217.

${\left(4x-9\right)}^2$

219.

$2\left(4p^2+1\right)$

221.

$\left(5-2y\right)\left(25+10y+4y^2\right)$

223.

$5{\left(3n+2\right)}^2$

225.

$\left(x-5-y\right)\left(x-5+y\right)$

227.

$\left(3x+1\right)\left(3x^2+1\right)$

229.

Answers will vary.

231.

Answers will vary.

Section 6.4 Exercises

233.

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

235.

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

237.

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

239.

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

241.

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

243.

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

245.

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

247.

$3xy\left(x-3\right)\left(x^2+3x+9\right)$

249.

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

251.

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

253.

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

255.

$4\left(x+3\right)\left(x+7\right)$

257.

$4u^2\left(u+v\right)\left(u^2-uv+v^2\right)$

259.

prime

261.

$10\left(m-5\right)\left(m+5\right)\left(m^2+25\right)$

263.

$3y\left(3x+2\right)\left(4x-1\right)$

265.

$\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)$

267.

$\left(y+1\right)\left(y-1\right)\left(y^2-y+1\right)\left(y^2+y+1\right)$

269.

$\left(3x-y+7\right)\left(3x-y-7\right)$

271.

${\left(3x-2\right)}^2$

273.

Answers will vary.

275.

Answers will vary.

Section 6.5 Exercises

277.

$a=10\text{/}3,a=7\text{/}2$

279.

$m=0,m=5\text{/}12$

281.

$x=1\text{/}2$

283.

$a=\frac{\mathrm{-4}}{5},a=6$

285.

$m=5\text{/}4,m=3$

287.

$a=\mathrm{-1},a=0$

289.

$m=12\text{/}7,m=\mathrm{-12}\text{/}7$

291.

$y=\mathrm{-9}\text{/}4,y=9\text{/}4$

293.

$n=\mathrm{-6}\text{/}11,\text{n}=6\text{/}11$

295.

$x=2,x=\mathrm{-5}$

297.

$x=3\text{/}2,x=\mathrm{-1}$

299.

$x=2,x=\mathrm{-4}\text{/}3$

301.

$x=3\text{/}2$

303.

$x=2,x=\mathrm{-4}\text{/}3$

305.

$x=\mathrm{-3}\text{/}2,x=1\text{/}3$

307.

$p=0,p=\mathrm{¾}$

309.

$x=0,x=6$

311.

$x=0,x=\mathrm{–1}\text{/}3$

313.

ⓐ $x=2$ or $x=6$ ⓑ $\left(2,\mathrm{-4}\right)$ $\left(6,\mathrm{-4}\right)$

315.

ⓐ $x=\frac{3}{2}$ or $x=\frac{3}{4}$
ⓑ $\left(\frac{3}{2},\mathrm{-4}\right)$ $\left(\frac{3}{4},\mathrm{-4}\right)$

317.

ⓐ $x=\frac{2}{3}$ or $x=-\frac{2}{3}$
ⓑ $\left(\frac{2}{3},0\right)$, $\left(-\frac{2}{3},0\right)$ ⓒ $\left(0,\mathrm{-4}\right)$

319.

ⓐ $x=\frac{5}{3}$ or $x=-\frac{1}{2}$
ⓑ $\left(\frac{5}{3},0\right)$, $\left(-\frac{1}{2},0\right)$ ⓒ $\left(0,\mathrm{-5}\right)$

321.

$\mathrm{-13},\mathrm{-11}$ and 11, 13

323.

$\mathrm{-14},\mathrm{-12}$ 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.

$3a{b}^2$

339.

$3y$

341.

$7(5y+12)$

343.

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

345.

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

347.

$\mathrm{-3}x\left(x^2-9x+4\right)$

349.

$\left(a+b\right)\left(x-y\right)$

351.

$\left(x-3\right)\left(x+7\right)$

353.

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

355.

$\left(a+3\right)\left(a+11\right)$

357.

$\left(m+9\right)\left(m-6\right)$

359.

$\left(x+5y\right)\left(x+7y\right)$

361.

$\left(a+7b\right)\left(a-3b\right)$

363.

Prime

365.

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

367.

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

369.

$\left(5y+1\right)\left(2y-11\right)$

371.

$\mathrm{-9}\left(9a+1\right)\left(a-2\right)$

373.

$\left(3a-1\right)\left(6a-1\right)$

375.

Prime

377.

$3\left(x+4\right)\left(x-3\right)$

379.

$3\left(2a-7\right)\left(3a+1\right)$

381.

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

383.

${\left(5x+3\right)}^2$

385.

$10{\left(2x+9\right)}^2$

387.

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

389.

$\left(13m+n\right)\left(13m-n\right)$

391.

$\left(3+11y\right)\left(3-11y\right)$

393.

$n\left(13n+1\right)\left(13n-1\right)$

395.

$6\left(4p^2+9\right)$

397.

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

399.

$\left(a+3-3b\right)\left(a+3+3b\right)$

401.

$\left(a-5\right)\left(a^2+5a+25\right)$

403.

$2\left(m+3\right)\left(m^2-3m+9\right)$

405.

$4x^2\left(6x+11\right)$

407.

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

409.

$5u^2\left(u+3\right)\left(u-3\right)$

411.

prime

413.

$\left(b-4\right)\left(b^2+4b+16\right)$

415.

$\left(2b+5c\right)\left(b-c\right)$

417.

$5\left(q+3\right)\left(q-6\right)$

419.

$10\left(m-5\right)\left(m+5\right)\left(m^2+25\right)$

421.

$\left(4x-3y+8\right)\left(4x-3y-8\right)$

423.

$b=\mathrm{-1}\text{/}5,b=\mathrm{-1}\text{/}6$

425.

$x=1\text{/}2$

427.

$x=\mathrm{-4},x=\mathrm{-5}$

429.

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

431.

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

433.

$x=2,x=\mathrm{-5}$

435.

$p=0,p=\mathrm{¾}$

437.

ⓐ $x=\mathrm{-7}$ or $x=\mathrm{-4}$
ⓑ $\left(\mathrm{-7},\mathrm{-8}\right)$ $\left(\mathrm{-4},\mathrm{-8}\right)$

439.

ⓐ $x=\frac{7}{8}$ or $x=-\frac{7}{8}$
ⓑ $\left(\frac{7}{8},0\right),$ $\left(-\frac{7}{8},0\right)$ ⓒ $\left(0,\mathrm{-49}\right)$

441.

The numbers are $\mathrm{-21}$ and $\mathrm{-19}$ or 19 and 21.

443.

The lengths are 8, 15, and 17 ft.

Practice Test

445.

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

447.

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

449.

$\left(x-8\right)\left(y+7\right)$

451.

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

453.

$3\left(x+5y\right)\left(x-5y\right)$

455.

$\left(x+5\right)\left(x^2-5x+25\right)$

457.

$\left(3x^2-5\right)\left(2x^2-3\right)$

459.

$a=4\text{/}5,a=\mathrm{-6}$

461.

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

463.

ⓐ $x=3$ or $x=4$ ⓑ $\left(3,\mathrm{-7}\right)$ $\left(4,\mathrm{-7}\right)$