 Intermediate Algebra 2e

# Chapter 6

### Be Prepared

6.1

$2·2·2·72·2·2·7$

6.2

72

6.3

$−21a2−24ab−21a2−24ab$

6.4

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

6.5

$6y2+23y+206y2+23y+20$

6.6

−54, 54

6.7

$27x627x6$

6.8

$m2+8m+16m2+8m+16$

6.9

$x2−9x2−9$

6.10

$y=35y=35$

6.11

$n(n−11)(n+2)n(n−11)(n+2)$

6.12

$8;x=28;x=2$

### Try It

6.1

$5m25m2$

6.2

$7x7x$

6.3

$3y2(3x+2x2+7y)3y2(3x+2x2+7y)$

6.4

$3p(p2−2pq+3q2)3p(p2−2pq+3q2)$

6.5

$2x2(x+6)2x2(x+6)$

6.6

$3y2(2y−5)3y2(2y−5)$

6.7

$3xy(5x2−xy+2y2)3xy(5x2−xy+2y2)$

6.8

$2ab(4a2+ab−3b2)2ab(4a2+ab−3b2)$

6.9

$−4b(b2−4b+2)−4b(b2−4b+2)$

6.10

$−7a(a2−3a+2)−7a(a2−3a+2)$

6.11

$(m+3)(4m−7)(m+3)(4m−7)$

6.12

$(n−4)(8n+5)(n−4)(8n+5)$

6.13

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

6.14

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

6.15

$(x−5)(x+2)(x−5)(x+2)$
$(5x−4)(4x−3)(5x−4)(4x−3)$

6.16

$(y+4)(y−7)(y+4)(y−7)$
$(7m−3)(6m−5)(7m−3)(6m−5)$

6.17

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

6.18

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

6.19

$(u−3)(u−6)(u−3)(u−6)$

6.20

$(y−7)(y−9)(y−7)(y−9)$

6.21

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

6.22

$(n−3)(n−4)(n−3)(n−4)$

6.23

$(a−b)(a−10b)(a−b)(a−10b)$

6.24

$(m−n)(m−12n)(m−n)(m−12n)$

6.25

prime

6.26

prime

6.27

$5x(x−1)(x+4)5x(x−1)(x+4)$

6.28

$6y(y−2)(y+5)6y(y−2)(y+5)$

6.29

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

6.30

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

6.31

$(2x−3)(4x−1)(2x−3)(4x−1)$

6.32

$(2y−7)(5y−1)(2y−7)(5y−1)$

6.33

$(3x+2y)(6x−5y)(3x+2y)(6x−5y)$

6.34

$(3x+y)(10x−21y)(3x+y)(10x−21y)$

6.35

$5n(n−4)(3n−5)5n(n−4)(3n−5)$

6.36

$8q(q+6)(7q−2)8q(q+6)(7q−2)$

6.37

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

6.38

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

6.39

$4(2x−3)(2x−1)4(2x−3)(2x−1)$

6.40

$3(3w−2)(2w−3)3(3w−2)(2w−3)$

6.41

$(h2−2)(h2+6)(h2−2)(h2+6)$

6.42

$(y2+4)(y2−5)(y2+4)(y2−5)$

6.43

$(x−3)(x−1)(x−3)(x−1)$

6.44

$(y−1)(y+1)(y−1)(y+1)$

6.45

$(2x+3)2(2x+3)2$

6.46

$(3y+4)2(3y+4)2$

6.47

$(8y−5)2(8y−5)2$

6.48

$(4z−9)2(4z−9)2$

6.49

$(7x+6y)2(7x+6y)2$

6.50

$(8m+7n)2(8m+7n)2$

6.51

$2y(2x−3)22y(2x−3)2$

6.52

$3q(3p+5)23q(3p+5)2$

6.53

$(11m−1)(11m+1)(11m−1)(11m+1)$

6.54

$(9y−1)(9y+1)(9y−1)(9y+1)$

6.55

$(14m−5n)(14m+5n)(14m−5n)(14m+5n)$

6.56

$(11p−3q)(11p+3q)(11p−3q)(11p+3q)$

6.57

$2y2(x−2)(x+2)(x2+4)2y2(x−2)(x+2)(x2+4)$

6.58

$7c2(a−b)(a+b)(a2+b2)7c2(a−b)(a+b)(a2+b2)$

6.59

$(x−5−y)(x−5+y)(x−5−y)(x−5+y)$

6.60

$(x+3−2y)(x+3+2y)(x+3−2y)(x+3+2y)$

6.61

$(x+3)(x2−3x+9)(x+3)(x2−3x+9)$

6.62

$(y+2)(y2−2y+4)(y+2)(y2−2y+4)$

6.63

$(2x−3y)(4x2+6xy+9y2)(2x−3y)(4x2+6xy+9y2)$

6.64

$125(2m−n)(4m2+2mn+n2)125(2m−n)(4m2+2mn+n2)$

6.65

$4(5p+q)(25p2−5pq+q2)4(5p+q)(25p2−5pq+q2)$

6.66

$2(6c+7d)(36c2−42cd+49d2)2(6c+7d)(36c2−42cd+49d2)$

6.67

$(−2y+1)(13y2+5y+1)(−2y+1)(13y2+5y+1)$

6.68

$(−4n+3)(31n2+21n+9)(−4n+3)(31n2+21n+9)$

6.69

$8y(y−1)(y+3)8y(y−1)(y+3)$

6.70

$5y(y−9)(y+6)5y(y−9)(y+6)$

6.71

$4x(2x−3)(2x+3)4x(2x−3)(2x+3)$

6.72

$3(3y−4)(3y+4)3(3y−4)(3y+4)$

6.73

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

6.74

$(3x−4y)2(3x−4y)2$

6.75

$2xy(25x2+36)2xy(25x2+36)$

6.76

$3xy(9y2+16)3xy(9y2+16)$

6.77

$2(5m+6n)(25m2−30mn+36n2)2(5m+6n)(25m2−30mn+36n2)$

6.78

$2(p+3q)(p2−3pq+9q2)2(p+3q)(p2−3pq+9q2)$

6.79

$4ab(a2+4)(a−2)(a+2)4ab(a2+4)(a−2)(a+2)$

6.80

$7xy(y2+1)(y−1)(y+1)7xy(y2+1)(y−1)(y+1)$

6.81

$6(x+b)(x−2c)6(x+b)(x−2c)$

6.82

$2(4x−1)(2x+3y)2(4x−1)(2x+3y)$

6.83

$4q(p−3)(p−1)4q(p−3)(p−1)$

6.84

$3p(2q+1)(q−2)3p(2q+1)(q−2)$

6.85

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

6.86

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

6.87

$m=23,m=−12m=23,m=−12$

6.88

$p=−34,p=34p=−34,p=34$

6.89

$c=2,c=43c=2,c=43$

6.90

$d=3,d=−12d=3,d=−12$

6.91

$p=75,p=−75p=75,p=−75$

6.92

$x=116,x=−116x=116,x=−116$

6.93

$m=1,m=32m=1,m=32$

6.94

$k=3,k=−3k=3,k=−3$

6.95

$a=−52,a=23a=−52,a=23$

6.96

$b=−2,b=−120b=−2,b=−120$

6.97

$x=0,x=32x=0,x=32$

6.98

$y=0,y=14y=0,y=14$

6.99

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

6.100

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

6.101

$x=1x=1$ or $x=52x=52$
$(1,0),(1,0),$ $(52,0)(52,0)$ $(0,5)(0,5)$

6.102

$x=−3x=−3$ or $x=56x=56$
$(−3,0),(−3,0),$ $(56,0)(56,0)$ $(0,−15)(0,−15)$

6.103

$−15,−17−15,−17$ and 15, 17

6.104

$−23,−21−23,−21$ 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

1.

$2pq2pq$

3.

$6m2n36m2n3$

5.

$2a2a$

7.

$5x3y5x3y$

9.

$3(2m+3)3(2m+3)$

11.

$9(n−7)9(n−7)$

13.

$3(x2+2x−3)3(x2+2x−3)$

15.

$2(4p2+2p+1)2(4p2+2p+1)$

17.

$8y2(y+2)8y2(y+2)$

19.

$5x(x2−3x+4)5x(x2−3x+4)$

21.

$3x(8x2−4x+5)3x(8x2−4x+5)$

23.

$6y2(2x+3x2−5y)6y2(2x+3x2−5y)$

25.

$4xy(5x2−xy+3y2)4xy(5x2−xy+3y2)$

27.

$−2(x+2)−2(x+2)$

29.

$−2x(x2−9x+4)−2x(x2−9x+4)$

31.

$−4pq(p2+3pq−4q)−4pq(p2+3pq−4q)$

33.

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

35.

$(b−2)(3b−13)(b−2)(3b−13)$

37.

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

39.

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

41.

$(u+2)(v−9)(u+2)(v−9)$

43.

$(u−1)(u+6)(u−1)(u+6)$

45.

$(3p−5)(3p+4)(3p−5)(3p+4)$

47.

$(n−6)(m−4)(n−6)(m−4)$

49.

$(x−7)(2x−5)(x−7)(2x−5)$

51.

$−9xy(2y+3x)−9xy(2y+3x)$

53.

$(x2+2)(3x−7)(x2+2)(3x−7)$

55.

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

57.

59.

### Section 6.2 Exercises

61.

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

63.

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

65.

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

67.

$(x−2)(x−6)(x−2)(x−6)$

69.

$(y−3)(y−15)(y−3)(y−15)$

71.

$(x−1)(x−7)(x−1)(x−7)$

73.

$(p−1)(p+6)(p−1)(p+6)$

75.

$(x−4)(x−2)(x−4)(x−2)$

77.

$(x−12)(x+1)(x−12)(x+1)$

79.

$(x+8y)(x−10y)(x+8y)(x−10y)$

81.

$(m+n)(m−65n)(m+n)(m−65n)$

83.

$(a+8b)(a−3b)(a+8b)(a−3b)$

85.

Prime

87.

Prime

89.

$p(p−10)(p+2)p(p−10)(p+2)$

91.

$3m(m−5)(m−2)3m(m−5)(m−2)$

93.

$5x2(x−3)(x+5)5x2(x−3)(x+5)$

95.

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

97.

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

99.

$(4w−1)(w−1)(4w−1)(w−1)$

101.

$(4q+1)(q−2)(4q+1)(q−2)$

103.

$(2p−5q)(3p−2q)(2p−5q)(3p−2q)$

105.

$(4a−3b)(a+5b)(4a−3b)(a+5b)$

107.

$−16(x+1)(x+1)−16(x+1)(x+1)$

109.

$−10q(3q+2)(q+4)−10q(3q+2)(q+4)$

111.

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

113.

$(2k−3)(2k−5)(2k−3)(2k−5)$

115.

$(3y+5)(2y−3)(3y+5)(2y−3)$

117.

$(2n+3)(n−15)(2n+3)(n−15)$

119.

$10(6y−1)(y+5)10(6y−1)(y+5)$

121.

$3z(8z+3)(2z−5)3z(8z+3)(2z−5)$

123.

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

125.

$12(4y−3)(y+1)12(4y−3)(y+1)$

127.

$(x2+1)(x2−7)(x2+1)(x2−7)$

129.

$(x2−7)(x2+4)(x2−7)(x2+4)$

131.

$(x−12)(x+1)(x−12)(x+1)$

133.

$(3y−4)(3y−1)(3y−4)(3y−1)$

135.

$(u−6)(u−6)(u−6)(u−6)$

137.

$(r−4s)(r−16s)(r−4s)(r−16s)$

139.

$(4y−7)(3y−2)(4y−7)(3y−2)$

141.

$(2n−1)(3n+4)(2n−1)(3n+4)$

143.

$13(z2+3z−2)13(z2+3z−2)$

145.

$3p(p+7)3p(p+7)$

147.

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

149.

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

151.

$(x2+2)(x2−6)(x2+2)(x2−6)$

153.

$(x−9)(x+6)(x−9)(x+6)$

155.

157.

### Section 6.3 Exercises

159.

$(4y+3)2(4y+3)2$

161.

$(6s+7)2(6s+7)2$

163.

$(10x−1)2(10x−1)2$

165.

$(5n−12)2(5n−12)2$

167.

$(7x+2y)2(7x+2y)2$

169.

$(10y−1)2(10y−1)2$

171.

$10j(k+4)210j(k+4)2$

173.

$3u2(5u−v)23u2(5u−v)2$

175.

$(5v−1)(5v+1)(5v−1)(5v+1)$

177.

$(2−7x)(2+7x)(2−7x)(2+7x)$

179.

$6p2(q−3)(q+3)6p2(q−3)(q+3)$

181.

$6(4p2+9)6(4p2+9)$

183.

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

185.

$(13c−6d)(13c+6d)(13c−6d)(13c+6d)$

187.

$(2z−1)(2z+1)(4z2+1)(2z−1)(2z+1)(4z2+1)$

189.

$2b2(3a−2)(3a+2)(9a2+4)2b2(3a−2)(3a+2)(9a2+4)$

191.

$(x−8−y)(x−8+y)(x−8−y)(x−8+y)$

193.

$(a+3−3b)(a+3+3b)(a+3−3b)(a+3+3b)$

195.

$(x+5)(x2−5x+25)(x+5)(x2−5x+25)$

197.

$(z2−3)(z4+3z2+9)(z2−3)(z4+3z2+9)$

199.

$(2−7t)(4+14t+49t2)(2−7t)(4+14t+49t2)$

201.

$(2y−5z)(4y2+10yz+25z2)(2y−5z)(4y2+10yz+25z2)$

203.

$(6a+5b)(36a2−30ab+25b2)(6a+5b)(36a2−30ab+25b2)$

205.

$7(k+2)(k2−2k+4)7(k+2)(k2−2k+4)$

207.

$2x2(1−2y)(1+2y+4y2)2x2(1−2y)(1+2y+4y2)$

209.

$9(x+1)(x2+3)9(x+1)(x2+3)$

211.

$−(3y+5)(21y2−30y+25)−(3y+5)(21y2−30y+25)$

213.

$(8a−5)(8a+5)(8a−5)(8a+5)$

215.

$3(3q−1)(3q+1)3(3q−1)(3q+1)$

217.

$(4x−9)2(4x−9)2$

219.

$2(4p2+1)2(4p2+1)$

221.

$(5−2y)(25+10y+4y2)(5−2y)(25+10y+4y2)$

223.

$5(3n+2)25(3n+2)2$

225.

$(x−5−y)(x−5+y)(x−5−y)(x−5+y)$

227.

$(3x+1)(3x2+1)(3x+1)(3x2+1)$

229.

231.

### Section 6.4 Exercises

233.

$(2n−1)(n+7)(2n−1)(n+7)$

235.

$a3(a2+9)a3(a2+9)$

237.

$(11r−s)(11r+s)(11r−s)(11r+s)$

239.

$8(m−2)(m+2)8(m−2)(m+2)$

241.

$(5w−6)2(5w−6)2$

243.

$(m+7n)2(m+7n)2$

245.

$7(b+3)(b−2)7(b+3)(b−2)$

247.

$3xy(x−3)(x2+3x+9)3xy(x−3)(x2+3x+9)$

249.

$(k−2)(k+2)(k2+4)(k−2)(k+2)(k2+4)$

251.

$5xy2(x2+4)(x+2)(x−2)5xy2(x2+4)(x+2)(x−2)$

253.

$3(5p+4)(q−1)3(5p+4)(q−1)$

255.

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

257.

$4u2(u+v)(u2−uv+v2)4u2(u+v)(u2−uv+v2)$

259.

prime

261.

$10(m−5)(m+5)(m2+25)10(m−5)(m+5)(m2+25)$

263.

$3y(3x+2)(4x−1)3y(3x+2)(4x−1)$

265.

$(2x−3y)(4x2+6xy+9y2)(2x−3y)(4x2+6xy+9y2)$

267.

$(y+1)(y−1)(y2−y+1)(y2+y+1)(y+1)(y−1)(y2−y+1)(y2+y+1)$

269.

$(3x−y+7)(3x−y−7)(3x−y+7)(3x−y−7)$

271.

$(3x−2)2(3x−2)2$

273.

275.

### Section 6.5 Exercises

277.

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

279.

$m=0,m=5/12m=0,m=5/12$

281.

$x=1/2x=1/2$

283.

$a=−45,a=6a=−45,a=6$

285.

$m=5/4,m=3m=5/4,m=3$

287.

$a=−1,a=0a=−1,a=0$

289.

$m=12/7,m=−12/7m=12/7,m=−12/7$

291.

$y=−9/4,y=9/4y=−9/4,y=9/4$

293.

$n=−6/11,n=6/11n=−6/11,n=6/11$

295.

$x=2,x=−5x=2,x=−5$

297.

$x=3/2,x=−1x=3/2,x=−1$

299.

$x=2,x=−4/3x=2,x=−4/3$

301.

$x=3/2x=3/2$

303.

$x=2,x=−4/3x=2,x=−4/3$

305.

$x=−3/2,x=1/3x=−3/2,x=1/3$

307.

$p=0,p=¾p=0,p=¾$

309.

$x=0,x=6x=0,x=6$

311.

$x=0,x=–1/3x=0,x=–1/3$

313.

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

315.

$x=32x=32$ or $x=34x=34$
$(32,−4)(32,−4)$ $(34,−4)(34,−4)$

317.

$x=23x=23$ or $x=−23x=−23$
$(23,0)(23,0)$, $(−23,0)(−23,0)$ $(0,−4)(0,−4)$

319.

$x=53x=53$ or $x=−12x=−12$
$(53,0)(53,0)$, $(−12,0)(−12,0)$ $(0,−5)(0,−5)$

321.

$−13,−11−13,−11$ and 11, 13

323.

$−14,−12−14,−12$ 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.

### Review Exercises

337.

$3ab23ab2$

339.

$3y3y$

341.

$7(5y+12)7(5y+12)$

343.

$3x(6x2−5)3x(6x2−5)$

345.

$4x(x2−3x+4)4x(x2−3x+4)$

347.

$−3x(x2−9x+4)−3x(x2−9x+4)$

349.

$(a+b)(x−y)(a+b)(x−y)$

351.

$(x−3)(x+7)(x−3)(x+7)$

353.

$(m2+1)(m+1)(m2+1)(m+1)$

355.

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

357.

$(m+9)(m−6)(m+9)(m−6)$

359.

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

361.

$(a+7b)(a−3b)(a+7b)(a−3b)$

363.

Prime

365.

$3y(y−5)(y−2)3y(y−5)(y−2)$

367.

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

369.

$(5y+1)(2y−11)(5y+1)(2y−11)$

371.

$−9(9a+1)(a−2)−9(9a+1)(a−2)$

373.

$(3a−1)(6a−1)(3a−1)(6a−1)$

375.

Prime

377.

$3(x+4)(x−3)3(x+4)(x−3)$

379.

$3(2a−7)(3a+1)3(2a−7)(3a+1)$

381.

$(x2−15)(x2+2)(x2−15)(x2+2)$

383.

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

385.

$10(2x+9)210(2x+9)2$

387.

$3u2(5u−v)23u2(5u−v)2$

389.

$(13m+n)(13m−n)(13m+n)(13m−n)$

391.

$(3+11y)(3−11y)(3+11y)(3−11y)$

393.

$n(13n+1)(13n−1)n(13n+1)(13n−1)$

395.

$6(4p2+9)6(4p2+9)$

397.

$(2z−1)(2z+1)(4z2+1)(2z−1)(2z+1)(4z2+1)$

399.

$(a+3−3b)(a+3+3b)(a+3−3b)(a+3+3b)$

401.

$(a−5)(a2+5a+25)(a−5)(a2+5a+25)$

403.

$2(m+3)(m2−3m+9)2(m+3)(m2−3m+9)$

405.

$4x2(6x+11)4x2(6x+11)$

407.

$(4n−7m)2(4n−7m)2$

409.

$5u2(u+3)(u−3)5u2(u+3)(u−3)$

411.

prime

413.

$(b−4)(b2+4b+16)(b−4)(b2+4b+16)$

415.

$(2b+5c)(b−c)(2b+5c)(b−c)$

417.

$5(q+3)(q−6)5(q+3)(q−6)$

419.

$10(m−5)(m+5)(m2+25)10(m−5)(m+5)(m2+25)$

421.

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

423.

$b=−1/5,b=−1/6b=−1/5,b=−1/6$

425.

$x=1/2x=1/2$

427.

$x=−4,x=−5x=−4,x=−5$

429.

$p=−52,p=8p=−52,p=8$

431.

$m=512,m=−512m=512,m=−512$

433.

$x=2,x=−5x=2,x=−5$

435.

$p=0,p=¾p=0,p=¾$

437.

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

439.

$x=78x=78$ or $x=−78x=−78$
$(78,0),(78,0),$ $(−78,0)(−78,0)$ $(0,−49)(0,−49)$

441.

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

443.

The lengths are 8, 15, and 17 ft.

### Practice Test

445.

$40a2(2+3a)40a2(2+3a)$

447.

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

449.

$(x−8)(y+7)(x−8)(y+7)$

451.

$(3s−2)2(3s−2)2$

453.

$3(x+5y)(x−5y)3(x+5y)(x−5y)$

455.

$(x+5)(x2−5x+25)(x+5)(x2−5x+25)$

457.

$(3x2−5)(2x2−3)(3x2−5)(2x2−3)$

459.

$a=4/5,a=−6a=4/5,a=−6$

461.

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

463.

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

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