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Mathematics LibreTexts

13.1.5: Chapter 5

( \newcommand{\kernel}{\mathrm{null}\,}\)

Be Prepared

5.1

11x2+8x+6

5.2

3n+9

5.3

200

5.4

8

5.5

x3y

5.6

10.946

5.7

2x+6

5.8

81; 81; 81

5.9

21

5.10

3+xd

5.11

6y2

5.12

11a2+7a+5

Try It

5.1

monomial, 0
polynomial, 3 trinomial, 4
binomial, 2 monomial, 10

5.2

binomial, 3 trinomial, 3 monomial, 0 polynomial, 4 monomial, 7

5.3

21q2 13mn3

5.4

−7c2 −10y2z3

5.5

5y2+3z2
m2n28m2+4n2

5.6

−4m2+n2
pq26p5q2

5.7

8x211x+8

5.8

17y2+14y+1

5.9

x2+3x5

5.10

6b2+3

5.11

−5ab+7b2

5.12

7mn+4n2

5.13

5x25xy+5y2

5.14

7x26xy2y2

5.15

x3-y3

5.16

p33p2q+q3

5.17

18 50 −15

5.18

20 2 −4

5.19

The height is 150 feet.

5.20

The height is 31 feet.

5.21

(f+g)(x)=3x26x3 (f+g)(3)=6
(fg)(x)=x22x+9
(fg)(−2)=17

5.22

(f+g)(x)=6x2x+7 (f+g)(3)=58
(fg)(x)=4x27x9
(fg)(−2)=21

5.23

b17 43x 12p6
x18

5.24

x16 10x+1 12z8
b19

5.25

x5 69 1x4
11215

5.26

y6 108 1m8
1911

5.27

1 1

5.28

1 1

5.29

1z3 1107 p8 64

5.30

1n2 110,000 q7
16

5.31

8116 n2m2

5.32

12527 b4a4

5.33

1z9 1p3q3 12v5u

5.34

1c15 1r2s8 30d3c8

5.35

b35 512 a48

5.36

z54 349 q29

5.37

32w5x5 1 116b12
64a8

5.38

−27y3 1 116x8
8c12

5.39

p410000 n7m7
81a4b12c8 27x6y9

5.40

−8q3 x4w4 x2y69z4
8n6m6

5.41

81c24d30 1a18
y15

5.42

256a22b24 1p39
2x3y10

5.43

9.6×104 7.8×10−3

5.44

4.83×104
1.29×10−2

5.45

1,300 −0.00012

5.46

−950,000 0.075

5.47

−0.006 20,000

5.48

−0.009 400,000

5.49

−35y11 6a5b6

5.50

54b9 8r11s8

5.51

−15y324y2+21y
12x4y220x3y3+12x2y4

5.52

8x424x3+20x2
−18a5b+12a4b236a3b3

5.53

x2+17x+72
15c2+14c8

5.54

20x2+51x+27
30y23y6

5.55

x22x35
15x2+29x14

5.56

b2+3b18
16y220y50

5.57

x38x2+6x48
8a2b2+12ab20

5.58

y39y2+7y63
8x2y2+2xy15

5.59

15m251m+42

5.60

42b253b+15

5.61

y38y2+17y6
y38y2+17y6

5.62

2x3+5x27x+20
y38y2+17y6

5.63

x2+18x+81
4c24cd+d2

5.64

y2+22y+121
16x240xy+25y2

5.65

36x225
16p249q2

5.66

4x249 9x2y2

5.67

FOIL; 18b2+77b18
Binomial Squares; 81p272p+16
Binomial Squares; 49y2+14y+1
Product of Conjugates; 16r29

5.68

Binomial Squares; 36x2+84x+49 Product of Conjugates; 9x216 FOIL; 10x229x+10 Binomial Squares; 36n212n+1

5.69

(f·g)(x)=x37x2+13x15
(f·g)(2)=−9

5.70

(f·g)(x)=x3+x252x28
(f·g)(2)=−120

5.71

9a5b

5.72

−9dc4

5.73

4y27x4

5.74

58m5n3

5.75

−4a+2b

5.76

8a5b+6a3b2

5.77

y+7

5.78

m+5

5.79

x33x2+2x+1+3x+3

5.80

x33x22x13x+3

5.81

x2+4x+16

5.82

25x2+10x+4

5.83

3x2+4x2;2

5.84

4x23x+1;1

5.85

x34x2+5;0

5.86

x33x2+2;0

5.87

(fg)(x)=x8
(fg)(−3)=−11

5.88

(fg)(x)=x9
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.

12x2 5a

11.

6w 19x2y

13.

12x25a

15.

6w+19x2y

17.

−22b 16xy

19.

−10a+5b
pq24p3q2

21.

−4a2+b2
x2y3x+7xy2

23.

xy25x5y2
19y+5z

25.

x3y+a2b4a5ab2

27.

u2+4v2+12a+8b

29.

4a3b

31.

−17x6

33.

11y2+4y+11

35.

−3x2+17x1

37.

11x25x+5

39.

6a24a1

41.

2m27m+4

43.

11a+3

45.

12s216s+9

47.

3x2x+4

49.

w2+3w+4

51.

11w66

53.

10x27xy+6y2

55.

10m2+3mn8n2

57.

−3ab+3b2

59.

p36p2q+pq2+4q3

61.

x3+2x2y5xy2+y3

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)=7x2+4x+4 (f+g)(2)=40
(fg)(x)=−3x212x2
(fg)(−3)=7

75.


(f+g)(x)=6x3x29x+3
(f+g)(2)=29
(fg)(x)=x2+5x+3
(fg)(−3)=−21

77.

Answers will vary.

79.

Answers will vary.

Section 5.2 Exercises

81.

d9 414x 8y4 w6

83.

n31 3x+6 56w6
a16

85.

mx+3

87.

ya+b

89.

x15 59 1q18 110

91.

p14 412 1b8 145

93.

1 1

95.

−1 −1

97.

1a2 11000 c5 9

99.

1r3 1100,000 q10
1,000

101.

6425 a2b2

103.

72964 v5u5

105.

125 125 25 −25

107.

35 115

109.

1b4 w2x9 −12cd4

111.

1 1u4v5 36r2s5

113.

1p

115.

m8 1018 1x12

117.

y3x 5xy 1q48

119.

9x2y2 1 125x4
16y6

121.

−125a3b3 1 136x6 9y8

123.

p532 y6x6 8x3y6z3
16p6q4

125.

a481b4 16m225 a4c49b6 q8r8p2

127.

1125t8 1t19 y43x2

129.

16m8n22 4p6

131.

7n 17n 17n

133.

19p2 3p2 −3p2

135.

x14

137.

a36

139.

8m18

141.

1,000x6y3

143.

16a12b8

145.

827x6y3

147.

1,024a10

149.

25,000p24

151.

x18y18

153.

144m8n22

155.

45x3 48y4

157.

12r4 13x11

159.

1j3

161.

4000n12

163.

3.4×105 4.1×10−2

165.

1.29×106
1.03×10−6

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.

30x8 15x8y2

181.

72c5 13m5n5

183.

−5t315t2+90t
9sr527s2r4+45s3r39sr527s2r4+45s3r3

185.

−5m315m2+90m−5m315m2+90m
−21x4y230x3y3+3x2y4−21x4y230x3y3+3x2y4

187.

y 2 + 12 y + 27 y 2 + 12 y + 27

189.

21 q 2 44 q 32 21 q 2 44 q 32

191.

y 2 8 y + 12 y 2 8 y + 12

193.

6 p 2 + 11 p + 5 6 p 2 + 11 p + 5

195.

m 2 + 7 m 44 m 2 + 7 m 44

197.

33 r 2 85 r 8 33 r 2 85 r 8

199.

y 3 + 3 y 2 4 y 12 y 3 + 3 y 2 4 y 12

201.

6 x 2 y 2 + 13 x y + 6 6 x 2 y 2 + 13 x y + 6

203.

y 4 11 y 2 + 28 y 4 11 y 2 + 28

205.

9 r 2 s 2 33 r s + 28 9 r 2 s 2 33 r s + 28

207.

u 3 + 7 u 2 + 14 u + 8 u 3 + 7 u 2 + 14 u + 8

209.

3 a 3 + 31 a 2 + 5 a 50 3 a 3 + 31 a 2 + 5 a 50

211.

6 a 4 13 a 3 + 15 a 2 + 35 a 50 6 a 4 13 a 3 + 15 a 2 + 35 a 50

213.

p 3 10 p 2 + 33 p 36 p 3 10 p 2 + 33 p 36

215.

6 r 3 41 r 2 61 r 9 6 r 3 41 r 2 61 r 9

217.

q 2 + 24 q + 144 q 2 + 24 q + 144

219.

4 y 2 12 y z + 9 z 2 4 y 2 12 y z + 9 z 2

221.

x 2 + 4 3 x + 4 9 x 2 + 4 3 x + 4 9

223.

1 64 x 2 1 36 x y + 1 81 y 2 1 64 x 2 1 36 x y + 1 81 y 2

225.

25 u 4 + 90 u 2 + 81 25 u 4 + 90 u 2 + 81

227.

64 p 6 48 p 3 + 9 64 p 6 48 p 3 + 9

229.

64j216 64 j 2 16

231.

81 c 2 25 81 c 2 25

233.

49 w 2 100 x 2 49 w 2 100 x 2

235.

p 2 16 25 q 2 p 2 16 25 q 2

237.

x2y281 x 2 y 2 81

239.

225 m 4 64 n 8 225 m 4 64 n 8

241.

t 2 18 t + 81 t 2 18 t + 81

243.

2 x 2 3 x y 2 y 2 2 x 2 3 x y 2 y 2

245.

9 p 2 64 9 p 2 64

247.

k 2 12 k + 36 k 2 12 k + 36

249.

8 x 3 x 2 y 2 + 64 x y 8 y 3 8 x 3 x 2 y 2 + 64 x y 8 y 3

251.

y 8 + 4 y 4 z + 4 z 2 y 8 + 4 y 4 z + 4 z 2

253.

m 6 16 m 3 n + 64 n 2 m 6 16 m 3 n + 64 n 2

255.

r 5 + r 2 s 2 r 3 s 3 s 5 r 5 + r 2 s 2 r 3 s 3 s 5

257.

18 p 9 18 p 9

259.

−10 j 15 −10 j 15

261.

9 d 7 9 d 7

263.

42 m 5 n 8 42 m 5 n 8

265.

5 q 5 10 q 4 + 30 q 3 5 q 5 10 q 4 + 30 q 3

267.

y 3 y 2 2 y y 3 y 2 2 y

269.

6 k 3 + 11 k 2 26 k + 4 6 k 3 + 11 k 2 26 k + 4

271.

121 b 2 121 b 2

273.

4 x 4 9 y 8 4 x 4 9 y 8

275.

9 d 2 + 6 d + 1 9 d 2 + 6 d + 1

277.

9 z 2 + 6 5 z + 1 25 9 z 2 + 6 5 z + 1 25

279.


(f·g)(x)=4x3x28x+5(f·g)(x)=4x3x28x+5
(f·g)(−2)=−15(f·g)(−2)=−15

281.

(f·g)(x)=49x264(f·g)(x)=49x264
(f·g)(−2)=132(f·g)(−2)=132

283.


(f·g)(x)=x4+6x3+9x2+10x12(f·g)(x)=x4+6x3+9x2+10x12 (f·g)(1)=14(f·g)(1)=14

285.

Answers will vary.

287.

Answers will vary.

Section 5.4 Exercises

289.

2 m 3 3 n 5 2 m 3 3 n 5

291.

−3 y 3 4 x 3 −3 y 3 4 x 3

293.

−3 q 5 p 5 −3 q 5 p 5

295.

5 v 4 u 2 5 v 4 u 2

297.

4 x 2 + 3 x 4 x 2 + 3 x

299.

−6 y 2 + 3 y −6 y 2 + 3 y

301.

6 r 3 + 11 r 2 s 8 r s 3 6 r 3 + 11 r 2 s 8 r s 3

303.

−5 y 3 + 1 4 y −5 y 3 + 1 4 y

305.

a 7 a 7

307.

4 x + 3 4 x + 3

309.

p + 3 8 p + 8 p + 3 8 p + 8

311.

2 n 2 6 n + 8 + 4 n + 3 2 n 2 6 n + 8 + 4 n + 3

313.

m 2 10 m + 100 m 2 10 m + 100

315.

25 y 2 + 20 y + 16 25 y 2 + 20 y + 16

317.

x 2 5 x + 6 ; 0 x 2 5 x + 6 ; 0

319.

2 x 2 3 x + 4 ; 4 2 x 2 3 x + 4 ; 4

321.

x 3 2 x 2 + 5 x 4 ; 2 x 3 2 x 2 + 5 x 4 ; 2

323.

3 x 3 2 x 2 4 x 2 ; 0 3 x 3 2 x 2 4 x 2 ; 0

325.

(fg)(x)=x6(fg)(x)=x6
(fg)(−5)=−11(fg)(−5)=−11

327.

(fg)(x)=x2+5x4(fg)(x)=x2+5x4
(fg)(0)=−4(fg)(0)=−4

329.


(f·g)(x)=x4+6x3+9x2+10x12;(f·g)(x)=x4+6x3+9x2+10x12; (f·g)(1)=14(f·g)(1)=14

331.

−9 −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 −13 y 3

349.

6 m 2 + 19 m 4 6 m 2 + 19 m 4

351.

5 u 2 + 4 u + 7 5 u 2 + 4 u + 7

353.

14 x 7 14 x 7

355.

−7 b 3 a −7 b 3 a

357.

8 a 2 5 a 2 8 a 2 5 a 2

359.

3 m 2 3 m + 3 3 m 2 3 m + 3

361.

3 y 2 8 y 14 3 y 2 8 y 14

363.

x 3 + 2 x 2 y 5 x y 2 + y 3 x 3 + 2 x 2 y 5 x y 2 + y 3

365.

165 39 5

367.

The height is 64feet64feet.

369.

(f+g)(x)=4x25x2(f+g)(x)=4x25x2 (f+g)(3)=19(f+g)(3)=19
(fg)(x)=−3x12(fg)(x)=−3x12
(fg)(−2)=−6(fg)(−2)=−6

371.

p 13 p 13

373.

a 6 a 6

375.

y a + b y a + b

377.

a 5 a 5

379.

1 x 5 1 x 5

381.

1

383.

−3 −3

385.

1 1000 1 1000

387.

1 8 n 1 8 n

389.

1 1000 1 1000

391.

36

393.

−125 −125

395.

1 125 1 125

397.

x 3 27 x 3 27

399.

3 10 3 10

401.

x 6 x 6

403.

1 u 3 v 5 1 u 3 v 5

405.

1 p 1 p

407.

1 q 16 1 q 16

409.

n 2 n 2

411.

1

413.

9 y 8 9 y 8

415.

81 x 4 y 8 z 4 81 x 4 y 8 z 4

417.

27 x 7 y 17 27 x 7 y 17

419.

3 y 4 4 x 4 3 y 4 4 x 4

421.

5.3 × 10 6 5.3 × 10 6

423.

29,000 29,000

425.

0.00009413 0.00009413

427.

0.00072 0.00072

429.

9,000 9,000

431.

10 c 10 10 c 10

433.

m 7 n 10 9 m 7 n 10 9

435.

a 4 9 a 3 36 a 2 a 4 9 a 3 36 a 2

437.

8 n 4 10 n 3 8 n 4 10 n 3

439.

y 2 + 8 y 48 y 2 + 8 y 48

441.

18 p 2 93 p + 110 18 p 2 93 p + 110

443.

k 2 3 k 54 k 2 3 k 54

445.

10 y 2 59 y + 63 10 y 2 59 y + 63

447.

x 2 + x 72 x 2 + x 72

449.

30 a 2 33 a + 3 30 a 2 33 a + 3

451.

15 b 3 b 2 47 b + 18 15 b 3 b 2 47 b + 18

453.

24 y 3 54 y 2 + 32 y 5 24 y 3 54 y 2 + 32 y 5

455.

x 2 + 3 2 x + 9 16 x 2 + 3 2 x + 9 16

457.

25 p 2 + 70 p q + 49 q 2 25 p 2 + 70 p q + 49 q 2

459.

36 x 2 y 2 36 x 2 y 2

461.

144 x 6 49 y 4 144 x 6 49 y 4

463.

9 p 9 9 p 9

465.

3 y 4 3 y 4

467.

4 a 5 b 2 4 a 5 b 2

469.

4 m 9 n 4 4 m 9 n 4

471.

−9 y 2 + 4 y −9 y 2 + 4 y

473.

−3 x 1 + 3 4 x −3 x 1 + 3 4 x

475.

y 3 + 33 y + 5 y 3 + 33 y + 5

477.

a 2 a + 1 2 a + 1 a 2 a + 1 2 a + 1

479.

2 x 2 5 x 4 ; 0 2 x 2 5 x 4 ; 0

481.

(fg)(x)=x6(fg)(x)=x6
(fg)(−2)=−8(fg)(−2)=−8

483.

−9 −9

485.

no

Practice Test

487.

trinomial 4

489.

6 x 2 3 x + 11 6 x 2 3 x + 11

491.

x x

493.

1 1

495.

1 8 y 3 1 8 y 3

497.

x 9 x 9

499.

4 r s 6 4 r s 6

501.

−48 x 5 y 9 −48 x 5 y 9

503.

7 m 2 + 19 m 6 7 m 2 + 19 m 6

505.

16 x 2 24 x + 9 16 x 2 24 x + 9

507.

3 y 2 7 x 3 y 2 7 x

509.

yes

511.

4.8 × 10 3 4.8 × 10 3

513.

3636 2121 99

515.

(fg)(x)=3x+4(fg)(x)=3x+4
(fg)(3)=13(fg)(3)=13


13.1.5: Chapter 5 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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