Solutions Polynomials EP 18-19.pdf

EP 2.1

1. Terms: 1 Coefficient: 1 Variable: 𝑦 Degree: 4 2. Terms: 3

Coefficient: 8, −1,81 Variable: 𝑐

Degree: 3 3. Terms: 4

Coefficient: 12,9, −1 Variable: 𝑧

Degree: 5 4. Terms: 4

Coefficient: −5,1,5, −1 Variable: 𝑚

Degree: 10 5. (student answer 6. No

7. Yes, degree 0 8. No

9. Yes, Degree 5 10. Student work 11. Student work 12. Student work 13.No

14. 𝑓(𝑥) = 3𝑥5 _{− 𝑥}3_{+ 5𝑥 − 2 ,}

𝑑𝑒𝑔𝑟𝑒𝑒 = 5

𝑎𝑛𝑑 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 3 15.𝑅(𝑥) = 30𝑥7+1

9𝑥

5_{− 13𝑥 +}

8 , 𝑑𝑒𝑔𝑟𝑒𝑒 = 7 𝑎𝑛𝑑

𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 30 16.No

17. No

18. 𝑟(𝑥) = 10𝑥4+ 4𝑥3− 13𝑥 − 7 , 𝑑𝑒𝑔𝑟𝑒𝑒 = 4 𝑎𝑛𝑑

𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = 10 19.9𝑥2 + 12𝑥

20. 3𝑥2_{+ 5𝑥 + 8}

21. 3𝑥3+ 6𝑥2+ 2𝑥 + 8

EP 2.2

1.

4𝑥

+ 3𝑥

+ 5𝑥

+ 15

2.

−2𝑥

− 1𝑥

− 2𝑥

3.

6𝑥

+ 7𝑥

− 12𝑥 + 1

4.

𝑥

+ 4𝑥

+ 1𝑥 + 1

5.

−3ℎ

+ 5ℎ

− 17ℎ − 25

6.

11𝑥

+ 𝑥

− 1𝑥 + 3

7.

−4𝑥

+ 𝑥

− 1𝑥 + 2

8.

−𝑥

+ 22𝑥

− 3𝑥 + 13

9.

−10𝑧

+ 7𝑧

− 3𝑧

−

13𝑧 + 23

10.

−7𝑥

+ 3𝑥

+ 4𝑥 − 8

11.

𝑥

− 13𝑥 + 8

12.

14𝑦

+ 26𝑦

− 3𝑦 −

33

13.

17𝑥

− 13𝑥

−

11𝑥

+ 16

14.

5𝑥

− 8𝑥

− 8𝑥 −

23

15.

5𝑥

− 𝑥 + 19

16.

13𝑥

− 12𝑥

+ 9𝑥

+

20

17.

−3𝑥

− 1𝑥 + 2

18.

3𝑥

− 10𝑥 + 4

19.

40𝑥

+ 10

20.

𝐺(𝑛) = 3𝑛

− 0.5𝑛

−

4𝑛 − 5

EP 2.3

1. 3𝑥3+ 21𝑥2+ 24𝑥

2. 18𝑥5_{+ 48𝑥}4_{− 78𝑥}3

3. −30𝑥2_{− 21𝑥 + 18}

4. −11𝑥3− 3𝑥2+ 77𝑥 + 21

5. 4𝑎2− 3𝑎𝑏 − 10𝑏2

6. 25𝑥6− 10𝑥3+ 1

7. 6𝑥3_{+ 40𝑥}2_{− 62𝑥 + 16}

8. −3𝑥7_{+ 5𝑥}6_{− 𝑥}5 _{+ 18𝑥}4 ₋

30𝑥3+ 6𝑥2

9. 20𝑥7− 52𝑥6+ 6𝑥5+ 12𝑥4

10. 54𝑥3_{− 18𝑥}2_{− 21𝑥 + 6}

11. −20𝑥5_{− 12𝑥}4_{− 54𝑥}3₋

23𝑥2− 39𝑥 − 6

12. 𝑥4 + 2𝑥3− 2𝑥2+ 7𝑥 − 2

13. 3𝑦4_{+ 𝑦}3_{− 13𝑦}2_{+ 9𝑦 − 4}

14. 6𝑥4_{+ 𝑥}3_{+ 44𝑥 − 21}

15. 5𝑥3− 6𝑥2− 5𝑥 + 6

16. 2𝑥4− 54𝑥2 + 100

17. 125𝑥3− 225𝑥2+ 135𝑥 − 27

18. −6𝑥3_{− 11𝑥}2_{+ 4𝑥 + 4}

19. 6𝑥3_{+ 22𝑥}3 _{+ 10𝑥 − 6}

20. 6𝑥3− 7𝑥2𝑦 − 9𝑥𝑦2− 2𝑦3

EP 2.4

1. −3𝑥2 _{+ 3𝑥 + 3}

2. −4𝑥2 _{+ 𝑥 − 4}

3. −5𝑥2 _{+ 3𝑥 − 9}

4. 𝑥5− 3𝑥3− 5𝑥

5. 10𝑥3− 2𝑥2− 3𝑥

6. 4𝑥 + 3

7. −10𝑥2_{+ 11𝑥}

8. 2𝑥2_{+ 5𝑥 + 2}

9. 6𝑥4− 15𝑥3+ 7𝑥2− 5

10. 52𝑥2− 22𝑥 − 20

11. 4𝑥3− 18𝑥2− 21𝑥 + 11

12. 20𝑥4_{− 27𝑥}3_{+ 29𝑥}2₊

EP 2.5R

1. (𝒙 + 𝟏𝟑)(𝒙 − 𝟏)

2. (𝒙 + 𝟒)(𝒙 + 𝟐)

3. (𝒙 − 𝟏𝟒)(𝒙 + 𝟐)

4. (𝒙 − 𝟖)(𝒙 − 𝟏𝟎)

5. (𝒙 − 𝟕)(𝒙 + 𝟐)

6. (𝒙 − 𝟏𝟖)(𝒙 + 𝟐)

7. (𝒙 − 𝟖)(𝒙 − 𝟐)

8. (𝒙 − 𝟔)(𝒙 + 𝟏)

9. (𝒙 − 𝟕)(𝒙 − 𝟏)

10. (𝒙 + 𝟔)(𝒙 + 𝟐)

11. (𝒙 + 𝟖)(𝒙 − 𝟑)

12. (𝒙 + 𝟔)(𝒙 − 𝟐)

13. (𝒙 + 𝟗)(𝒙 + 𝟓)

14. (𝒙 + 𝟑)(𝒙 − 𝟓)

15. (𝒙 − 𝟗)(𝒙 − 𝟒)

16. (𝒙 − 𝟑)(𝒙 − 𝟏𝟐)

17. (𝒙 + 𝟐𝟕)(𝒙 + 𝟑)

18. (𝒙 − 𝟐)(𝟓𝒙 + 𝟑)

19. (𝟐𝒙 + 𝟓)(𝒙 + 𝟐)

20. (𝒙 − 𝟐)(𝟑𝒙 + 𝟒)

21. 𝟑𝒙(𝒙 − 𝟐)(𝒙 + 𝟏)

22. 𝟐𝒙𝟑_(𝒙𝟐_{− 𝟒𝒙 + 𝟓)}

23. 𝟒(𝒙 − 𝟗)(𝒙 − 𝟏)

24. 𝟐(𝒙 − 𝟖)(𝒙 + 𝟑)

25. 𝒙(𝒙 − 𝟔)

26. −𝟑(𝒙 + 𝟒)(𝒙 − 𝟏)

27. 𝟑(𝒙 + 𝟒)(𝒙 − 𝟐)

28. 𝒙(𝒙 + 𝟕)

29. −𝟗𝒙𝟐_(𝒙𝟐_{− 𝟓𝒙 + 𝟏)}

EP 2.6

On Worksheet

EP 2.7

1. (𝒙𝟐_{+ 𝟑)(𝒙 − 𝟐)}

2. (𝒙 + 𝟐)(𝒙 − 𝟐)(𝒙 − 𝟏)

3. (𝒙 + 𝟑)(𝒙 − 𝟑)(𝒙 − 𝟔)

4. (2𝑥2_{+ 2)(3𝑥 + 1)}

5. (𝑥 + 5)(2𝑥2_{− 3)}

6. (𝑥3_{+ 1)(4𝑥 + 7)}

7. (8𝑥2_{+ 1)(𝑥 − 8)}

8. (2𝑥3_{− 5)(6𝑥 + 1)}

9. (5𝑥 − 1)(4𝑥2_{+ 3)}

10. 4𝑥(𝑦 + 3)(𝑥 + 4)

11. (4𝑦 + 1)(7𝑦2_{− 4)}

12. (5𝑥2_{+ 6)(5𝑥 + 1)}

13. (4𝑥2_{− 3)(7𝑥 + 4)}

14. (8𝑥2_{− 1)(3𝑥 − 8)}

15. (𝑥2_{− 5)(𝑥 − 3)}

16. (𝑥2_{+ 3)(4𝑥 − 7)}

17. 𝑥(𝑥 + 1)(𝑥 − 1)(𝑥 − 3)

18. 𝑥(𝑥 + 3)(𝑥 − 3)(2𝑥 − 1)

EP 2.8

1. (𝑥2 _{− 11)(𝑥}2_{− 3)}

2. (𝑥2 _{+ 4)(𝑥}2_{+ 3)}

3. (𝑥2 _{− 9)(𝑥}2_{+ 4)}

4. (𝑥2 _{− 6)(𝑥}2_{− 1)}

5. (𝑥2 _{− 3)(𝑥}2_{− 1)}

6. (𝑥2 _{− 15)(𝑥}2_{− 2)}

7. (𝑥2 _{+ 12)(𝑥}2_{+ 5)}

8. (𝑥2 _{− 3)(𝑥}2_{+ 2)}

9. (𝑥2 _{− 8)(𝑥}2_{+ 4)}

10.(𝑥2 _{+ 5)(𝑥}2_{+ 5)}

11.(𝑥2 _{− 8)(𝑥}2_{− 1)}

12. (𝑥2_{− 9)(𝑥}2_{− 5)}

13. (𝑥2_{− 8)(𝑥}2_{+ 3)}

14. (𝑥2_{+ 15)(𝑥}2_{− 2)}

15. (𝑥2_{− 9)(𝑥}2_{+ 2)}

16. (𝑥2_{− 8𝑦)(𝑥}2_{− 5𝑦)}

17. (𝑥2_{− 9)(𝑥}2_{+ 9)}

18. (𝑥2_{− 7)(𝑥}2_{+ 7)}

19. (𝑥5_{− 12)(𝑥}5_{+ 12)}

20. (5𝑥2_{− 4)(5𝑥}2_{+ 4)}

21. (3𝑥2_{− 2𝑦)(3𝑥}2_{+ 2𝑦)}

22. (9𝑥2_{− 10)(9𝑥}2_{+ 10)}

23. (𝑥2_{− 6𝑦)(𝑥}2_{+ 6𝑦)}

24. (𝑥3_{− 10)(𝑥}3_{+ 10)}

EP 2.9

1. (𝑥 + 2𝑖)(𝑥 − 2𝑖)

2. (𝑥 + 15𝑖)(𝑥 − 15𝑖)

3. (𝑥 + 10𝑖)(𝑥 − 10𝑖)

4. (𝑥 + 8𝑖)(𝑥 − 8𝑖)

5. (𝑥 + 11𝑖)(𝑥 − 11𝑖)

6. (𝑥 + 14𝑖)(𝑥 − 14𝑖)

7. (𝑥 + 5𝑖)(𝑥 − 5𝑖)

8. (𝑥 + 9𝑖)(𝑥 − 9𝑖)

9. (𝑥 + 7𝑖)(𝑥 − 7𝑖)

10. (𝑥 + 6𝑖)(𝑥 − 6𝑖)

11. (𝑥 + 12𝑖)(𝑥 − 12𝑖)

12. (𝑥 + 4𝑖)(𝑥 − 4𝑖)

13. (𝑥 + 13𝑖)(𝑥 − 13𝑖)

14. (𝑥 + 3𝑖)(𝑥 − 3𝑖)

15.(𝑥 + 16𝑖)(𝑥 − 16𝑖)

16. (𝑥 + 𝑦𝑖)(𝑥 − 𝑦𝑖)

17. (5𝑥 + 22𝑖)(5𝑥 − 22𝑖)

EP 2.10

1. 𝑥 → ∞, 𝑦 → ∞ 𝑥 → −∞, 𝑦 → −∞

2. 𝑥 → ∞, 𝑦 → ∞ 𝑥 → −∞, 𝑦 → ∞

3. 𝑥 → ∞, 𝑦 → ∞ 𝑥 → −∞, 𝑦 → −∞

4. 𝑥 → ∞, 𝑦 → −∞ 𝑥 → −∞, 𝑦 → +∞

5. 𝑥 → ∞, 𝑦 → −∞ 𝑥 → −∞, 𝑦 → −∞

6. 𝑥 → ∞, 𝑦 → +∞ 𝑥 → −∞, 𝑦 → +∞

7.

𝑥 → ∞, 𝑦 → +∞ 𝑥 → −∞, 𝑦 → +∞

8.

𝑥 → ∞, 𝑦 → +∞ 𝑥 → −∞, 𝑦 → −∞

9.

𝑥 → ∞, 𝑦 → +∞ 𝑥 → −∞, 𝑦 → +∞

10.

𝑥 → ∞, 𝑦 → −∞ 𝑥 → −∞, 𝑦 → +∞

11.

𝑥 → ∞, 𝑦 → −∞ 𝑥 → −∞, 𝑦 → −∞

12.

𝑥 → ∞, 𝑦 → +∞ 𝑥 → −∞, 𝑦 → −∞

13. 𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → +∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → −∞,

14. 𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → −∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → −∞,

15. 𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → −∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

16. 𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → +∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

17. 𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → −∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → −∞,

18. 𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → −∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

19. 4 possible solutions

𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → +∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

20. 3 possible solutions

𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → +∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

21. 5 possible solutions

𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → −∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

22. 4 possible solutions

𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → +∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → +∞,

23. 6 possible solutions

𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → −∞

𝑙𝑒𝑓𝑡: 𝑥 → −∞, 𝑦 → −∞,

24. 7 possible solutions

𝑟𝑖𝑔ℎ𝑡 𝑥 → +∞, 𝑦 → +∞

EP 2.11 Honors/Reg

1. a) 4, 1

b) 𝑥 → −∞, 𝑦 → ∞ 𝑥 → ∞, 𝑦 → ∞ c) (−2,0) (2,0)

(0,16) d) 4 e) 3 f) (0,16)

(−2,0) (2,0) g) D: (−∞, ∞)

R: 𝑦 ≥ 0

h) 𝐼: 𝑥 > 2, −2 < 𝑥 < 0 𝐷: 𝑥 < −2, 0 < 𝑥 < 2 2.

a) 3, 1

b) 𝑥 → −∞, 𝑦 → −∞ 𝑥 → ∞, 𝑦 → ∞ c) (−1,0) (2,0)

(0, −2) d) 3 e) 2 f) (−1,0)

(−1, −4) a) D: (−∞, ∞)

R: (−∞, ∞)

g) 𝐼: 𝑥 < −1 𝑎𝑛𝑑 𝑥 < 1 𝐷: − 1 < 𝑥 < 1 3.

a) D: (−∞, ∞) R: 𝑦 ≥ 0

b) 𝐼: − 2 < 𝑥 < −1 𝑎𝑛𝑑 𝑥 < 0

𝐷: 𝑥 < −2 𝑎𝑛𝑑 −1 < 𝑥 < 0 4.

b) 3,-1

c) 𝑥 → −∞, 𝑦 → ∞ 𝑥 → ∞, 𝑦 → −∞ d) (0,0) (2,0) (4,0)

(0,0)

e) 3 f) 2

g) (3.1,3.1) (0.8, −3.1) h) D: (−∞, ∞)

R: (−∞, ∞) i) 𝐼: 0.8 < 𝑥 < 3.1

𝐷: 𝑥 < 0.8 𝑎𝑛𝑑 𝑥 > 3.1 5.

a) 4, -2

b) 𝑥 → −∞, 𝑦 → −∞ 𝑥 → ∞, 𝑦 → −∞ c) (−2,0) (2,0)

(0,4) d) 4 e) 3

f) (−1.2, 8.5) (1.2,8.5) (0,4)

c) D: (−∞, ∞) R: 𝑦 ≤ 8.5

g) 𝐼: 𝑥 < −1.2, 0 < 𝑥 < 1.2 𝐷: − 1.2 < 𝑥 < 0 𝑎𝑛𝑑 𝑥 > 1.2 6.

a) D: (−∞, ∞) R: (−∞, ∞)

b) 𝐼: − 1.6 < 𝑥 < 0.9 𝐷: 𝑥 < −1.6, 𝑥 > 0.9

EP 2.12

1.

2.

3.

4.

5.

6.

7. 𝑓(𝑥) = 𝑥4− 2𝑥3− 3𝑥2, Reason

8. 𝑓(𝑥) = −0.25(𝑥 + 2)(𝑥 − 5)(𝑥 + 3) +

2, reason

9. 𝑓(𝑥) = 2𝑥6_{− 13𝑥}5_{+ 26𝑥}4₋

7𝑥3_{− 28𝑥}2_{+ 20𝑥 + 20, reason}

10. 𝑓(𝑥) = 3𝑥5_{+ 20𝑥}4_{− 10𝑥}3₋

240𝑥2_{− 250𝑥 + 200, reason}

11.𝑓(𝑥) = (𝑥 + 3)3_{, reason}

12.𝑓(𝑥) = 𝑥4− 4𝑥3− 2𝑥2+ 12𝑥 − 3, reason

EP 2.13

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

EP 2.14

Stations in Classroom

EP 2.15

1. # of zeros: 3

Real Zeros: 1, 𝑥 = 0

# of Imaginary Zeros: 2 2. # of zeros: 4

Real Zeros: 2, 𝑥 = −2,2

# of Imaginary Zeros: 2 3. # of zeros: 4

Real Zeros: 4,

𝑥 = 3,1, −1, −3

# of Imaginary Zeros: 0 4. # of zeros: 3

Real Zeros: 1, 𝑥 = −5

# of Imaginary Zeros: 2 5. # of zeros: 3

Real Zeros: 1, 𝑥 = 3

# of Imaginary Zeros: 2 6. # of zeros: 3

Real Zeros: 3,

𝑥 = 0,5 + √5

2 ,

5 − √5 2

# of Imaginary Zeros: 0 7. # of zeros: 7

Real Zeros: 4,

𝑥 = 2, −2, √2, −√2

# of Imaginary Zeros: 0 8. # of zeros: 4

Real Zeros: 2, 𝑥 = 0,3

# of Imaginary Zeros: 2 9. # of zeros: 4

Real Zeros: 2, 𝑥 = −2,2

# of Imaginary Zeros: 2 10.# of zeros: 3

Real Zeros: 3,

𝑥 = 0,3 + 3√5

2 ,

3 − 3√5 2

# of Imaginary Zeros: 0 11.# of zeros: 3

Real Zeros: 1, 𝑥 = −6

# of Imaginary Zeros: 2 12.# of zeros: 3

Real Zeros: 3,

𝑥 = 0, −3, 1

# of Imaginary Zeros: 0

EP 2.16

1. # of Zeros: 3, 𝑥 = 4, 5𝑖, −5𝑖

2. # of Zeros: 4,

𝑥 = 𝑖√5, −𝑖√5, 𝑖√3, −𝑖√3

3. # of Zeros: 4,

𝑥 = √5, −√5, 3, −3

4. # of Zeros:3, 𝑥 =3

4, 2𝑖, −2𝑖

5. # of Zeros: 4, 𝑥 = 0, 0, 3, −6

6. # of Zeros: 4,

𝑥 = 2𝑖, −2𝑖, 𝑖√2, − 𝑖√2

7. # of Zeros: 3, 𝑥 =6

5, 3, −3

8. # of Zeros: 3, 𝑥 = 3, 2, −2

9. # of Zeros: 4,

𝑥 = 2√2, −2√2, 2𝑖√2, −2𝑖√2

10. # of Zeros: 3, 𝑥 = 0, −1, −1

5

11. # of Zeros: 4,

𝑥 = 3, 0, 1𝑖, −1𝑖

12. # of Zeros: 4,

EP 2.17

1. # of Zeros: 3, 𝑥 = 0, 0, 3

2. # of Zeros: 4,

𝑥 = 0,2, −1,1, −1

3. # of Zeros: 3, 𝑥 = 3,1

2, − 1 2

4. # of Zeros: 3, 𝑥 = −1,1, −3

5. # of Zeros: 3,

𝑥 = 2, √3, −√3

6. # of Zeros: 4,

𝑥 = 3, −3,2𝑖, −2𝑖

7. # of Zeros: 4,

𝑥 = −4, 𝑖√7, −𝑖√7

8. # of Zeros: 4,

𝑥 = √3, −√3, 2√2𝑖, −2√2𝑖

9. # of Zeros: 3,

𝑥 = 2, −2, −1

10. # of Zeros: 3, 𝑥 = 0, −4, −4

11. # of Zeros: 3, 𝑥 = 3, 0, 0

12. # of Zeros: 5,

𝑥 = 0, 2, −2, √2, −√2

EP 2.18

1. 𝑓(𝑥) = 𝑥3 − 4𝑥

2. 𝑓(𝑥) = 𝑥4 − 13𝑥2 + 36

3. 𝑓(𝑥) = 𝑥3_{+ 2𝑥}2_{− 1𝑥 − 2}

4. 𝑓(𝑥) = −𝑥3_{+ 6𝑥}2_{− 32}

5. 𝑓(𝑥) = −𝑥3− 3𝑥2+ 4

6. 𝑓(𝑥) = 𝑥5− 5𝑥3+ 4𝑥

7. 𝑓(𝑥) = −𝑥4+ 9𝑥2

8. 𝑓(𝑥) = −𝑥4_{+ 3𝑥}3_{+ 5𝑥}2₋

3𝑥 − 4

9. 𝑓(𝑥) = 𝑥4+ 4𝑥3+ 4𝑥2

10. 𝑓(𝑥) = −𝑥3+ 𝑥2+ 2𝑥

11. 𝑓(𝑥) = 𝑥4− 2𝑥3

12. 𝑓(𝑥) = −𝑥3_{+ 7𝑥}2_{− 14𝑥 + 8}

EP 2.19

1. −√2

2. 1 + √3

3. −2𝑖

4. 1 − 5𝑖

5. 𝑎 − √𝑏

6. 𝑎 − 𝑏𝑖

7. 3 + √2

8. 1 + 2𝑖

9. 1 − √5

10. 2 − 𝑖

11. 2 + √3

12. 3 + 4𝑖

13. −√5, +√13

14. 4 − √6, −√3

15.1 + √10 , 2 − 𝑖

16. 1 − 𝑖, +5𝑖

17. 2 − 3𝑖, −6𝑖

18. 4 + 𝑖, 3 − 7𝑖

19. −3𝑖

20. 1 + 𝑖

21. −4𝑖

22. 3 − 𝑖

23. 2𝑖

24. −8𝑖

25. 𝑥2 + 9

26. 𝑥2+ 4

27. 𝑥2_{+ 16}

28. 5

29. 13

30. 𝑥2+ 64

31. 𝑥2 + 25

32. 𝑥2 _{− 2}

33. 𝑥2 _{− 7}

34. 401

35. 25

36. 49𝑥2 + 1

37. 81𝑥2 + 225

EP 2.20

1. 𝑓(𝑥) = 𝑥3+ 2𝑥2− 35𝑥

2. 𝑓(𝑥) = 𝑥3_{− 5𝑥}2_{+ 4𝑥 − 20} 3. 𝑓(𝑥) = 𝑥4_{+ 11𝑥}3_{+ 30𝑥}2_{− 32𝑥 − 160}

4. 𝑓(𝑥) = 𝑥3− 11𝑥2+ 39𝑥 − 45 5. 𝑓(𝑥) = 𝑥4_{− 10𝑥}2_{+ 9}

6. 𝑓(𝑥) = 𝑥3− 2𝑥2+ 9𝑥 − 18

7. 𝑓(𝑥) = 𝑥3− 2𝑥2− 2𝑥 + 4

8. 𝑓(𝑥) = 𝑥2+ 16

9. 𝑓(𝑥) = 𝑥3_{− 8𝑥}2_{+ 𝑥 + 42}

10. 𝑓(𝑥) = 𝑥3_{− 15𝑥}2_{− 16𝑥 + 240}

11. 𝑓(𝑥) = 𝑥3− 9𝑥2+ 4𝑥 − 36

12. 𝑓(𝑥) = 𝑥4_{+ 8𝑥}3_{− 8𝑥}2_{− 96𝑥 + 144}

13. 𝑓(𝑥) = 𝑥3− 3𝑥2− 10𝑥

14. 𝑓(𝑥) = 𝑥4_{− 24𝑥}2_{− 25}

EP 2.21

1. 𝑓(𝑥) = 𝑥3+ 5𝑥2+ 2𝑥 + 10

2. 𝑓(𝑥) = 𝑥4_{− 3𝑥}2_{+ 5𝑥}2_{− 27𝑥 − 36}

3. 𝑓(𝑥) = 𝑥3− 3𝑥2+ 49𝑥 − 147

4. 𝑓(𝑥) = 𝑥4_{− 𝑥}3_{+ 10𝑥}2_{− 16𝑥 − 96}

5. 𝑓(𝑥) = 𝑥3+ 2𝑥2+ 36𝑥 + 72

6. 𝑓(𝑥) = 𝑥4− 63𝑥2− 64

7. 𝑓(𝑥) = 𝑥3_{− 7𝑥}2_{+ 𝑥 − 7}

8. 𝑓(𝑥) = 𝑥4_{− 44𝑥}2_{− 245}

9. 𝑓(𝑥) = 𝑥3− 4𝑥2+ 9𝑥 − 10

EP 2.22

1. 𝑥2_{− 4𝑥 + 6}

2. 𝑥3_{+ 2𝑥}2 _{− 7𝑥 + 3}

3. 𝑥2− 3𝑥 + 10 − 14

𝑥+2

4. −4𝑥

5. 𝑥 + 10 + 47

𝑥−5

6. −2𝑥3 − 9𝑥2− 17𝑥 − 26 +

84 −2𝑥+3

7. 𝑥2− 3𝑥 + 9 − 36

𝑥+3

8. 2𝑥2_{+ 𝑥 + 5}

9. 𝑥2_{− 𝑥 − 6}

10. 6𝑥2_{− 12𝑥 + 25 −} 51 𝑥+2

11. −2𝑥3_{+ 2𝑥}2_{+ 4𝑥 + 11 +} 21

𝑥−2

12. 𝑥 − 7 + 11

𝑥+1

13. 2𝑥2 _{+ 𝑥 + 6 +} 5 𝑥−2

14. 3𝑥2_{+ 2𝑥 + 20 +} 58 𝑥−4

15. 2𝑥2− 10𝑥 + 15 − 41

𝑥+3

16. 𝑥 − 4 + 20

𝑥+4

17. 𝑥 + 3 + 4

𝑥−1

18. 𝑥2_{− 𝑥 + 9 −} 2 𝑥+1

19. 2𝑥3+ 3𝑥2+ 7𝑥 + 7 + 30

2𝑥−3

20. 2𝑥 − 2 + 7

5𝑥+2

EP 2.23

1. Divisor 𝑥2_{+ 2𝑥 + 1}

remainder is 10

2. 𝑄(𝑥) = 3𝑥2_{− 5𝑥 + 6}

3. 𝑝(𝑥) = 𝑥3− 4𝑥2+ 2𝑥 + 4

4. 6𝑥2− 5𝑥 − 5 ÷ 2𝑥 + 1

5. Divisor 2𝑥2 _{+ 𝑥 − 15}

remainder is zero

6. 𝑝(𝑥) = 𝑥4− 6𝑥3− 19𝑥2+ 24𝑥 7. 2𝑥3−𝑥2−13𝑥−6

𝑥−2

8. Divisor 𝑥 + 2

remainder is 30

9. 𝑥3_{+ 12𝑥}2_{+ 17𝑥 − 30 ÷ 𝑥 − 3}

10. 𝑥4+8𝑥3−3𝑥2−24𝑥

𝑥−3

11.𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵𝑎𝑠𝑒 = 𝑥2+ 5𝑥 + 6 𝑉(𝑥) = (𝑥 + 1)(𝑥2_{+ 5𝑥 + 6)}

𝑉(𝑥) = (𝑥 + 1)(𝑥 + 2)(𝑥 + 3) 12.𝐴𝑟𝑒𝑎 𝑜𝑓 𝐵𝑎𝑠𝑒 = 𝑥2− 2𝑥

𝑉(𝑥) = (𝑥 + 3)(𝑥2_{− 2𝑥)}

𝑉(𝑥) = (𝑥 + 3)𝑥(𝑥 − 2)

EP 2.24

1. No, not a factor 2. Yes, a factor 3. Yes is a factor 4. No, not a factor 5. No, not a factor 6. Yes, is a factor 7. No, not a factor 8. No, not a factor 9. Yes, is a factor 10. Remainder of −144

Divisor 𝑥 − 3

No, not a factor 11. Remainder of 60

Divisor 𝑥 + 3

No, not a factor 12. Remainder of −10

Divisor 𝑥 − 1

No, not a factor 13. Remainder of 7068

Divisor 𝑥 − 10

No, not a factor

EP 2.25

1. (𝑥 + 4)(𝑥2_{− 4𝑥 + 16)}

2. (𝑥 − 6)(𝑥2 _{+ 6𝑥 + 36)}

3. (5𝑦 + 3)(25𝑦2_{− 15𝑦 + 9)}

4. (3𝑥 + 1)(9𝑥2_{− 3𝑥 + 1)}

5. (4𝑦 − 3)(16𝑦2_{+ 12𝑦 + 9)}

6. (𝑥2 _{− 3)(𝑥}4_{+ 3𝑥}2 _{+ 9)}

7. (2𝑥 + 3)(4𝑥2_{− 6𝑥 + 9)}

8. (3𝑥 − 𝑦)(9𝑥2_{+ 3𝑥𝑦 + 𝑦}2₎

9. (𝑥 − 6𝑦)(𝑥2_{+ 6𝑥𝑦 + 36𝑦}2₎

10.(𝑥 − 8)(𝑥2_{+ 8𝑥 + 64)}

EP 2.26

10. a. Answers will vary b. Answers will vary 11. 9885

12. 0. 36̅̅̅̅ pounds consumed per

year

13. 1.3135 ppm/year increase

EP 2.27

1. 𝑉(𝑥) = 12𝑥2 _{+ 12𝑥 − 9}

2. 𝑉(𝑥) = 𝜋(𝑥3− 3𝑥2+ 8𝑥 + 4)

3. 𝑉(ℎ) = ℎ3 + 5ℎ2+ 4ℎ

4. 𝑉(𝑥) = 6𝑥3 − 17𝑥2 + 6𝑥 + 8

5. 𝑉(𝑥) = (𝑥 + 5)(𝑥 + 4)(𝑥 + 3)

𝑥 = 1

6. 𝑉(𝑥) = 2𝑥2 − 16𝑥 + 32

7. 𝑉(𝑙) = 𝑙(𝑙 − 1)(𝑙 − 2) = 60 𝑙 = 5

8. 𝑉(ℎ) = ℎ3 _{+ 10ℎ}2_{+ 21ℎ}

9. 𝑉(𝑥) = 𝜋(𝑥3_{− 6𝑥}2_{+ 9𝑥)}

10.𝑉(𝑥) = 𝑥3 − 8𝑥2

11. 𝑉(𝑥) = (𝑥 + 2)(𝑥 + 2)(𝑥 + 3)

𝑥 = 1.608

12. 𝑉(𝑥) = (𝑥 + 1)3 𝑥 = 7.434

13. A.

𝑉𝑇(𝑥) = (2𝑥 + 1)(𝑥 + 3)(𝑥 + 4)

𝑉𝐼(𝑥) = (2𝑥 + 1)(𝑥 + 2)(𝑥 + 1)

EP 2.28

1. 16𝑥4_{+ 96𝑥}3_{+ 216𝑥}2_{+ 216𝑥 + 81}

2. 𝑥5_{+ 15𝑥}4_{+ 90𝑥}3_{+ 270𝑥}2_{+ 405𝑥 + 243}

3. 81𝑥4_{+ 216𝑥}3_{𝑦 + 216𝑥}2_𝑦2_{+ 96𝑥𝑦}3_{+ 16𝑦}4 4. 8𝑥3 _{− 12𝑥}2 _{+ 6𝑥 − 1}

5. 160𝑥3

6. 90𝑥2

7. 90𝑥3

8. −4𝑥𝑦3

9. 80

10. 108

11. 90