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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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 3
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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 4
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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 5
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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 6
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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 7
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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 8
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}
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 9
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)
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 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
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 11
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)
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 12
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)
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 13
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}
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 14
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)
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 15
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)
Chapter 9 – Polynomials Answer Key
CK-12 Algebra I Concepts 16
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
7. The numbers are 3 and 5
8. The numbers are 6 and 2
9. The picture is 2ft by 2ft