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Chapter 9 – Polynomials Answer Key

CK-12 Algebra I Concepts 1

9.1 Polynomials in Standard Form

Answers

1. Not a polynomial (fractional exponent)

2. Polynomial

3. Not a polynomial (negative exponent)

4. Not a polynomial (2nd term equal to fractional exponent)

5. Not a polynomial (1st term equal to fractional exponent: √𝑥 = 𝑥1

2)

6. Polynomial (exponent equal to 3)

7. −2𝑥 + 3 degree 1

8. 3𝑥3 − 4𝑥 + 8 degree 3

9. 8𝑥3 − 5𝑥2 + 2𝑥 − 5 degree 3

10. −9𝑥4 + 𝑥2 + 12 degree 4

11. 2𝑥2 + 2𝑥 degree 2



9.2 Addition and Subtraction of Polynomials

Answers

1. (𝑥 + 8) + (−3𝑥 − 5) = −𝟐𝒙 + 𝟑

2. (−2𝑥2 + 4𝑥 − 12) + (7𝑥 + 𝑥2) = −𝒙𝟐 + 𝟏𝟏𝒙 − 𝟏𝟐

3. (2𝑎2𝑏 − 2𝑎 + 9) + (5𝑎2𝑏 − 4𝑏 + 5) = 𝟕𝒂𝟐𝒃 − 𝟐𝒂 − 𝟒𝒃 + 𝟏𝟒

4. 𝟔. 𝟗𝒂𝟐 + 𝟑. 𝟏𝒂 + 𝟐𝒂𝒃 − 𝟒. 𝟖𝒃𝟐 + 𝒃

5. (3

5𝑥2 −

1

4𝑥 + 4) + (

1

10𝑥2 +

1

2𝑥 − 2

1

5) =

𝟏

𝟐𝒙𝟐 +

𝟏

𝟒𝒙 + 𝟏

𝟒

𝟓

6. (−𝑡 + 5𝑡2) − (5𝑡2 + 2𝑡 − 9) = −𝟑𝒕 + 𝟗

7. (−𝑦2 + 4𝑦 − 5) − (5𝑦2 + 2𝑦 + 7) = −𝟔𝒚𝟐 + 𝟐𝒚 − 𝟏𝟐

8. (−5𝑚2 − 𝑚) − (3𝑚2 + 4𝑚 − 5) = −𝟖𝒎𝟐 − 𝟓𝒎 + 𝟓

9. 𝟑𝒂𝟐𝒃𝟐 − 𝟐𝒂𝟐𝒃 − 𝟑𝒂𝒃𝟐 + 𝟓𝒃𝟐

10. 𝟐. 𝟑𝒙𝟐𝒚 − 𝟓𝒙𝒚 + 𝟒𝒙 − 𝟐𝒚 + 𝟑

11. 𝑥𝑦 + 2𝑥𝑧

12. 4𝑎𝑏 + 𝑎2 OR 4𝑎𝑏 + 𝑎𝑐

13. 2𝑥𝑦 − 2𝑥2

14. 3𝑎𝑏



9.3 Multiplication of Monomials by Polynomials

Answers

1. (2𝑥)(−7𝑥) = −𝟏𝟒𝒙𝟐

2. (10𝑥)(3𝑥𝑦) = 𝟑𝟎𝒙𝟐𝒚

3. (4𝑚𝑛)(0.5𝑛𝑚2) = 𝟐𝒏𝟐𝒎𝟑

4. (−5𝑎2𝑏)(−12𝑎3𝑏3) = 𝟔𝟎𝒂𝟓𝒃𝟒

5. (3𝑥𝑦2𝑧2)(15𝑥2𝑦𝑧3) = 𝟒𝟓𝒙𝟑𝒚𝟑𝒛𝟓

6. 17(8𝑥 − 10) = 𝟏𝟑𝟒𝒙 − 𝟏𝟕𝟎

7. 2𝑥(4𝑥 − 5) = 𝟖𝒙𝟐 − 𝟏𝟎𝒙

8. 9𝑥3(3𝑥2 − 2𝑥 + 7) = 𝟐𝟕𝒙𝟓 − 𝟏𝟖𝒙𝟒 + 𝟔𝟑𝒙𝟑

9. 3𝑥(2𝑦2 + 𝑦 − 5) = 𝟔𝒙𝒚𝟐 + 𝟑𝒙𝒚 − 𝟏𝟓𝒙

10. 10𝑞(3𝑞2𝑟 + 5𝑟) = 𝟑𝟎𝒒𝟑𝒓 + 𝟓𝟎𝒒𝒓

11. −3𝑎2𝑏(9𝑎2 − 4𝑏2) = −𝟐𝟕𝒂𝟒𝒃 + 𝟏𝟐𝒂𝟐𝒃𝟑



9.4 Multiplication of Polynomials by Binomials

Answers

1. (𝑥 − 3)(𝑥 + 2) = 𝒙𝟐 − 𝒙 − 𝟔

2. (𝑎 + 𝑏)(𝑎 − 5) = 𝒂𝟐 − 𝟓𝒂 + 𝒃𝒂 − 𝟓𝒃

3. (𝑥 + 2)(𝑥2 − 3) = 𝒙𝟑 + 𝟐𝒙𝟐 − 𝟑𝒙 − 𝟔

4. (𝑎2 + 2)(3𝑎2 − 4) = 𝟑𝒂𝟒 + 𝟐𝒂𝟐 − 𝟖

5. (7𝑥 − 2)(9𝑥 − 5) = 𝟔𝟑𝒙𝟐 − 𝟓𝟑𝒙 + 𝟏𝟎

6. (2𝑥 − 1)(2𝑥2 − 𝑥 + 3) = 𝟒𝒙𝟑 − 𝟒𝒙𝟐 + 𝟕𝒙 − 𝟑

7. (3𝑥 + 2)(9𝑥2 − 6𝑥 + 4) = 𝟐𝟕𝒙𝟑 + 𝟖

8. (𝑎2 + 2𝑎 − 3)(𝑎2 − 3𝑎 + 4) = 𝒂𝟒 − 𝒂𝟑 − 𝟓𝒂𝟐 + 𝟏𝟕𝒂 − 𝟏𝟐

9. 3(𝑥 − 5)(2𝑥 + 7) = (3𝑥 − 15)(2𝑥 + 7) = 𝟔𝒙𝟐 − 𝟗𝒙 − 𝟏𝟎𝟓

10. 5𝑥(𝑥 + 4)(2𝑥 − 3) = 𝟏𝟎𝒙𝟑 + 𝟐𝟓𝒙𝟐 − 𝟔𝟎𝒙

11. (2𝑥 + 4)(𝑥 + 5) = 𝟐𝒙𝟐 + 𝟏𝟒𝒙 + 𝟐𝟎

12. 3𝑥2 + 8𝑥

13. (3𝑥 + 4)(2𝑥)(𝑥 + 1) = (6𝑥2 + 8𝑥)(𝑥 + 1) = 𝟔𝒙𝟑 + 𝟏𝟒𝒙𝟐 + 𝟖𝒙

14. 4𝑥(3𝑥 − 1)(2𝑥 + 4) = (12𝑥2 − 4𝑥)(2𝑥 + 4) = 𝟐𝟒𝒙𝟑 + 𝟒𝟎𝒙𝟐 − 𝟏𝟔𝒙



9.5 Special Products of Polynomials

Answers

1. 𝑥2 + 18𝑥 + 81

2. 9𝑥2 − 42𝑥 + 49

3. 25𝑥2 − 10𝑥𝑦 + 𝑦2

4. 4𝑥6 − 12𝑥3 + 9

5. 16𝑥4 + 8𝑥2𝑦2 + 𝑦4

6. 64𝑥2 − 48 + 9

7. 4𝑥2 + 20𝑥 + 25

8. 𝑥2𝑦2 − 2𝑥𝑦2 + 𝑦2

9. 4𝑥2 − 1

10. 𝑥2 − 144

11. 25𝑎2 − 4𝑏2

12. 𝑎2𝑏2 − 1

13. 𝑧4 − 𝑦2



14. 4𝑞6 − 𝑟4

15. 49𝑠2 − 𝑡2

16. 𝑥4𝑦4 − 𝑥2𝑦4

17. (𝑎 − 𝑏)2 OR 𝑎2 − 2𝑎𝑏 + 𝑏2

18. 2,475

19. 3,136

20. 999,996

21. 1,584

22. 99.75

23. 981.96

24. 9,975

25. –4



9.6 Monomial Factors of Polynomials

Answers

1. 𝑥

2. 3𝑥

3. 5𝑥4

4. 2𝑥

5. 2𝑥4

6. 12𝑥𝑦

7. 𝑎

8. 3

9. 2𝑎

10. 15𝑦10

11. 4𝑥𝑦

12. 2

13. 5𝑦



9.7 Zero Product Principle

Answers

1. 𝑥 = {12, 0}

2. 𝑥 = {−1

2,

1

2}

3. 𝑥 = {−7

2,

4

3, 5}

4. 𝑥 = {−9, 0,20

7}

5. 𝑥 = 0, 𝑦 = −3

6. 𝑥 = 0 or 𝑥 = 2𝑦

7. 𝑦 = {0, 6}

8. 𝑥 = {0, 3}

9. 𝑎 = {−1

4, 0}

10. 𝑏 = {0,5

3}

11. 𝑥 = {−3, 3}

12. 𝑥 = {0, 5}



9.8 Factorization of Quadratic Expressions

Answers

1. (𝑥 + 9)(𝑥 + 1)

2. (𝑥 + 5)(𝑥 + 10)

3. (𝑥 + 7)(𝑥 + 3)

4. (𝑥 + 4)(𝑥 + 12)

5. (𝑥 + 9)(𝑥 + 5)

6. (𝑥 + 25)(𝑥 + 2)

7. (𝑥 + 20)(𝑥 + 2)

8. (𝑥 + 7)(𝑥 + 8)

9. (𝑥 + 1)2

10. (𝑥 + 6)(𝑥 + 4)

11. (𝑥 + 9)(𝑥 + 8)

12. (𝑥 + 15)(𝑥 + 10)



9.9 Factorization of Quadratic Expressions with

Negative Coefficients

Answers

1. (𝑥 − 3)(𝑥 − 8)

2. (𝑥 − 6)(𝑥 − 7)

3. (𝑥 − 3)(𝑥 − 11)

4. (𝑥 − 4)(𝑥 − 5)

5. (𝑥 + 7)(𝑥 − 2)

6. (𝑥 + 9)(𝑥 − 3)

7. (𝑥 + 13)(𝑥 − 6)

8. (𝑥 + 8)(𝑥 − 4)

9. (𝑥 + 3)(𝑥 − 15)

10. (𝑥 + 5)(𝑥 − 10)

11. (𝑥 + 5)(𝑥 − 8)

12. (𝑥 + 7)(𝑥 − 8)

13. – (𝑥 + 1)2



14. – ((𝑥 + 8)(𝑥 − 3))

15. – ((𝑥 − 6)(𝑥 − 12))

16. – ((𝑥 − 10)(𝑥 − 15))

17. (𝑥 + 12)(𝑥 + 9)

18. – ((𝑥 − 5)(𝑥 − 6))

19. (𝑥 + 16)(𝑥 − 4)

20. (𝑥 + 3)(𝑥 − 20)

21. (𝑥 + 9)(𝑥 − 4)



9.10 Factorization using Difference of Squares

Answers

1. (𝑥 − 2)(𝑥 + 2)

2. (𝑥 − 6)(𝑥 + 6)

3. – (𝑥 − 10)(𝑥 + 10)

4. (𝑥 + 20)(𝑥 − 20)

5. (3𝑥 + 2)(3𝑥 − 2)

6. (5𝑥 + 7)(5𝑥 − 7)

7. (3𝑎 − 5𝑏)(3𝑎 + 5𝑏)

8. – (6𝑥 − 5)(6𝑥 + 5)

9. (2𝑥 − 𝑦)(2𝑥 + 𝑦)

10. (4𝑥 − 9𝑦)(4𝑥 + 9𝑦)



9.11 Factorization using Perfect Square Trinomials

Answers

1. (𝑥 + 4)2

2. (𝑥 − 9)2

3. −(𝑥 − 12)2

4. (𝑥 + 7)2

5. (2𝑥 − 1)2

6. (5𝑥 + 6)2

7. (2𝑥 − 3𝑦)2

8. (𝑥2 + 11)2

9. 𝑥 = {5, 6}

10. 𝑥 = {−7, 3}

11. 𝑥 = {7}

12. 𝑥 = {−8, 8}

13. 𝑥 = 12

14. 𝑥 = {±5

2}

15. 𝑥 = −13

16. 𝑥 = {−10, −6}



9.12 Factoring Completely

Answers

1. 2(𝑥 + 5)(𝑥 + 3)

2. 5(𝑥 − 7)(𝑥 − 7)

3. – (𝑥(𝑥 − 7)(𝑥 − 10))

4. 2(𝑥 − 4)(𝑥 + 4)(𝑥2 + 16)

5. 𝑥2(5𝑥 − 2)(5𝑥 − 2)

6. 3𝑥(2𝑥 + 1)(2𝑥 + 1)

7. 3(2𝑐 − 5)(2𝑐 + 5)

8. 6(𝑥 + 10)(𝑥 − 10)

9. −5(𝑡 + 2)(𝑡 + 2)

10. 6(𝑥 + 4)(𝑥 − 1)

11. – ((𝑛 − 3)(𝑛 − 7))

12. 2(𝑎 + 1)(𝑎 − 8)



9.13 Factoring by Grouping

Answers

1. (2𝑥 − 3)(3𝑥 + 5)

2. (5𝑥 + 1)(𝑥 − 7)

3. (9𝑥 − 1)(𝑥 − 1)

4. (4𝑥 − 5)(𝑥 + 8)

5. (2𝑎 + 3𝑏)(𝑎 − 3𝑏)

6. (𝑥 + 3)(5𝑥 − 2𝑦)

7. (4𝑥 − 3)(𝑥 + 7)

8. (6𝑥 + 1)(𝑥 + 1)

9. (2𝑥 − 1)(2𝑥 + 5)

10. (3𝑥 + 7)(𝑥 + 3)

11. 2(3𝑥 + 2)(𝑥 − 1)

12. (4𝑥 + 3)(2𝑥 − 5)



9.14 Solving Problems by Factoring

Answers

1. It is a 3-4-5 right triangle

2. 𝑆𝑖𝑑𝑒 1 = 5 (𝑥), 𝑠𝑖𝑑𝑒 𝑡𝑤𝑜 = 12(𝑥 + 7)

3. 𝑆𝑖𝑑𝑒 1 = 15, 𝑆𝑖𝑑𝑒 2 = 12

4. 𝑥 = 11

5. The numbers are 8 and 15

6. 14 feet by 34 feet



9. The picture is 2ft by 2ft