Answer Key
Chapter 9 – Polynomials
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 = 𝑥 2 )
6.
Polynomial (exponent equal to 3)
7.
−2𝑥 + 3 degree 1
8.
3𝑥 3 − 4𝑥 + 8 degree 3
9.
8𝑥 3 − 5𝑥 2 + 2𝑥 − 5 degree 3
1
10. −9𝑥 4 + 𝑥 2 + 12 degree 4
11. 2𝑥 2 + 2𝑥 degree 2
CK-12 Algebra I Concepts
1
Answer Key
Chapter 9 – Polynomials
9.2
Addition and Subtraction of Polynomials
Answers 1.
(𝑥 + 8) + (−3𝑥 − 5) = −𝟐𝒙 + 𝟑
2.
(−2𝑥 2 + 4𝑥 − 12) + (7𝑥 + 𝑥 2 ) = −𝒙𝟐 + 𝟏𝟏𝒙 − 𝟏𝟐
3.
(2𝑎2 𝑏 − 2𝑎 + 9) + (5𝑎2 𝑏 − 4𝑏 + 5) = 𝟕𝒂𝟐 𝒃 − 𝟐𝒂 − 𝟒𝒃 + 𝟏𝟒
4.
𝟔. 𝟗𝒂𝟐 + 𝟑. 𝟏𝒂 + 𝟐𝒂𝒃 − 𝟒. 𝟖𝒃𝟐 + 𝒃
5.
( 𝑥 2 − 𝑥 + 4) + (
6.
(−𝑡 + 5𝑡 2 ) − (5𝑡 2 + 2𝑡 − 9) = −𝟑𝒕 + 𝟗
7.
(−𝑦 2 + 4𝑦 − 5) − (5𝑦 2 + 2𝑦 + 7) = −𝟔𝒚𝟐 + 𝟐𝒚 − 𝟏𝟐
8.
(−5𝑚2 − 𝑚) − (3𝑚2 + 4𝑚 − 5) = −𝟖𝒎𝟐 − 𝟓𝒎 + 𝟓
9.
𝟑𝒂𝟐 𝒃𝟐 − 𝟐𝒂𝟐 𝒃 − 𝟑𝒂𝒃𝟐 + 𝟓𝒃𝟐
3 5
1 4
1 2 𝑥 10
1 2
1 5
𝟏 𝟐
𝟏 𝟒
+ 𝑥 − 2 ) = 𝒙𝟐 + 𝒙 + 𝟏
𝟒 𝟓
10. 𝟐. 𝟑𝒙𝟐 𝒚 − 𝟓𝒙𝒚 + 𝟒𝒙 − 𝟐𝒚 + 𝟑
11. 𝑥𝑦 + 2𝑥𝑧
12. 4𝑎𝑏 + 𝑎2 OR 4𝑎𝑏 + 𝑎𝑐
13. 2𝑥𝑦 − 2𝑥 2
14. 3𝑎𝑏
CK-12 Algebra I Concepts
2
Chapter 9 – Polynomials
9.3
Answer Key
Multiplication of Monomials by Polynomials
Answers 1.
(2𝑥)(−7𝑥) = −𝟏𝟒𝒙𝟐
2.
(10𝑥)(3𝑥𝑦) = 𝟑𝟎𝒙𝟐 𝒚
3.
(4𝑚𝑛)(0.5𝑛𝑚2 ) = 𝟐𝒏𝟐 𝒎𝟑
4.
(−5𝑎2 𝑏)(−12𝑎3 𝑏3 ) = 𝟔𝟎𝒂𝟓 𝒃𝟒
5.
(3𝑥𝑦 2 𝑧 2 )(15𝑥 2 𝑦𝑧 3 ) = 𝟒𝟓𝒙𝟑 𝒚𝟑 𝒛𝟓
6.
17(8𝑥 − 10) = 𝟏𝟑𝟒𝒙 − 𝟏𝟕𝟎
7.
2𝑥(4𝑥 − 5) = 𝟖𝒙𝟐 − 𝟏𝟎𝒙
8.
9𝑥 3 (3𝑥 2 − 2𝑥 + 7) = 𝟐𝟕𝒙𝟓 − 𝟏𝟖𝒙𝟒 + 𝟔𝟑𝒙𝟑
9.
3𝑥(2𝑦 2 + 𝑦 − 5) = 𝟔𝒙𝒚𝟐 + 𝟑𝒙𝒚 − 𝟏𝟓𝒙
10. 10𝑞(3𝑞 2 𝑟 + 5𝑟) = 𝟑𝟎𝒒𝟑 𝒓 + 𝟓𝟎𝒒𝒓
11. −3𝑎2 𝑏(9𝑎2 − 4𝑏 2 ) = −𝟐𝟕𝒂𝟒 𝒃 + 𝟏𝟐𝒂𝟐 𝒃𝟑
CK-12 Algebra I Concepts
3
Chapter 9 – Polynomials
9.4
Answer Key
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)(𝑥 2 − 3) = 𝒙𝟑 + 𝟐𝒙𝟐 − 𝟑𝒙 − 𝟔
4.
(𝑎2 + 2)(3𝑎2 − 4) = 𝟑𝒂𝟒 + 𝟐𝒂𝟐 − 𝟖
5.
(7𝑥 − 2)(9𝑥 − 5) = 𝟔𝟑𝒙𝟐 − 𝟓𝟑𝒙 + 𝟏𝟎
6.
(2𝑥 − 1)(2𝑥 2 − 𝑥 + 3) = 𝟒𝒙𝟑 − 𝟒𝒙𝟐 + 𝟕𝒙 − 𝟑
7.
(3𝑥 + 2)(9𝑥 2 − 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. 3𝑥 2 + 8𝑥
13. (3𝑥 + 4)(2𝑥)(𝑥 + 1) = (6𝑥 2 + 8𝑥)(𝑥 + 1) = 𝟔𝒙𝟑 + 𝟏𝟒𝒙𝟐 + 𝟖𝒙
14. 4𝑥(3𝑥 − 1)(2𝑥 + 4) = (12𝑥 2 − 4𝑥)(2𝑥 + 4) = 𝟐𝟒𝒙𝟑 + 𝟒𝟎𝒙𝟐 − 𝟏𝟔𝒙
CK-12 Algebra I Concepts
4
Chapter 9 – Polynomials
9.5
Answer Key
Special Products of Polynomials
Answers 1.
𝑥 2 + 18𝑥 + 81
2.
9𝑥 2 − 42𝑥 + 49
3.
25𝑥 2 − 10𝑥𝑦 + 𝑦 2
4.
4𝑥 6 − 12𝑥 3 + 9
5.
16𝑥 4 + 8𝑥 2 𝑦 2 + 𝑦 4
6.
64𝑥 2 − 48 + 9
7.
4𝑥 2 + 20𝑥 + 25
8.
𝑥 2 𝑦 2 − 2𝑥𝑦 2 + 𝑦 2
9.
4𝑥 2 − 1
10. 𝑥 2 − 144
11. 25𝑎2 − 4𝑏 2
12. 𝑎2 𝑏 2 − 1
13. 𝑧 4 − 𝑦 2
CK-12 Algebra I Concepts
5
Chapter 9 – Polynomials
Answer Key
14. 4𝑞6 − 𝑟 4 15. 49𝑠 2 − 𝑡 2
16. 𝑥 4 𝑦 4 − 𝑥 2 𝑦 4
17. (𝑎 − 𝑏)2 OR 𝑎2 − 2𝑎𝑏 + 𝑏 2
18. 2,475
19. 3,136
20. 999,996
21. 1,584
22. 99.75
23. 981.96
24. 9,975
25. –4
CK-12 Algebra I Concepts
6
Chapter 9 – Polynomials
9.6
Answer Key
Monomial Factors of Polynomials
Answers 1.
𝑥
2.
3𝑥
3.
5𝑥 4
4.
2𝑥
5.
2𝑥 4
6.
12𝑥𝑦
7.
𝑎
8.
3
9.
2𝑎
10. 15𝑦10
11. 4𝑥𝑦
12. 2
13. 5𝑦
CK-12 Algebra I Concepts
7
Chapter 9 – Polynomials
9.7
Answer Key
Zero Product Principle
Answers 1.
𝑥 = {12, 0}
2.
𝑥 = {− 2 , 2}
3.
𝑥 = {− 2 , 3 , 5}
4.
𝑥 = {−9, 0,
5.
𝑥 = 0, 𝑦 = −3
6.
𝑥 = 0 or 𝑥 = 2𝑦
7.
𝑦 = {0, 6}
8.
𝑥 = {0, 3}
9.
𝑎 = {− 4 , 0}
1 1
7 4
20 } 7
1
5
10. 𝑏 = {0, 3}
11. 𝑥 = {−3, 3}
12. 𝑥 = {0, 5}
CK-12 Algebra I Concepts
8
Chapter 9 – Polynomials
9.8
Answer Key
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)
CK-12 Algebra I Concepts
9
Chapter 9 – Polynomials
9.9
Answer Key
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
CK-12 Algebra I Concepts
10
Chapter 9 – Polynomials
Answer Key
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)
CK-12 Algebra I Concepts
11
Chapter 9 – Polynomials
9.10
Answer Key
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)
CK-12 Algebra I Concepts
12
Chapter 9 – Polynomials
9.11
Answer Key
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.
(𝑥 2 + 11)2
9.
𝑥 = {5, 6}
10. 𝑥 = {−7, 3} 11. 𝑥 = {7} 12. 𝑥 = {−8, 8} 13. 𝑥 = 12 5
14. 𝑥 = {± 2} 15. 𝑥 = −13 16. 𝑥 = {−10, −6}
CK-12 Algebra I Concepts
13
Chapter 9 – Polynomials
9.12
Answer Key
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)(𝑥 2 + 16)
5.
𝑥 2 (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)
CK-12 Algebra I Concepts
14
Chapter 9 – Polynomials
9.13
Answer Key
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)
CK-12 Algebra I Concepts
15
Chapter 9 – Polynomials
9.14
Answer Key
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
CK-12 Algebra I Concepts
16
