Chapter R: Basic Algebraic Concepts
R.1: Exponents and Polynomials
1. 1 42 3 1 42 2 1 42 32 1 42 5
2. 152 2 152 6 152 26 152 8
3. 20 1
4. 20 112 1
5. 15m2 0 1, if m 0
6. 1 4z2 0 1, if z 0
7. 122 2 5 225 210
8. 164 2 3 643 612
9. 12x 5y 4 2 3 23 1x 5 2 3 1y 4 2 3 23x 15y 12 or 8x 15y 12
10. 1 4m3n9 2 2 1 42 2 1m3 2 2 1n9 2 2 42m6n18 or 16m6n18
p4 2 p42 p8
11. a b a 2 b 2
q q q
r 8 3 r 83 r 24
12. a 2
b 23 6
s s s
13. 5x 11 is a polynomial. It is a monomial since it has one term. It has degree 11 since 11 is the highest exponent.
14. 9y 12 y 2 is a polynomial. It is a binomial since it has two terms. It has degree 12 since 12 is the highest exponent.
15. 18p5q 6pq is a polynomial. It is a binomial since it has two terms. It has degree 6 since 6 is the sum of the
exponents in the term 18p5q. (The term 6pq has degree 2.)
16. 2a6 5a2 4a is a polynomial. It is a trinomial since it has three terms. It has degree 6 since 6 is the highest
exponent.
17. 12 x 2 13 x 6 is a polynomial. It is a binomial since it has two terms. It has degree 6 since 6 is the highest
exponent.
18. 17 m5n2 213 m3n2 is a polynomial. It is a binomial since it has two terms. It has degree 7 since 7 is the
sum of the exponents in the term 17 m5n2. (The other term has degree 5.)
1 22 3 42
19. r s r s rs3 is a polynomial. It is a trinomial since it has three terms. It has degree 6 since 6 is the
3 5
3
sum of the exponents in the term r 4s 2. (The other terms have degree 4.)
5
5 2 5
20. 2 3 is not a polynomial since positive exponents in the denominator are equivalent to negative
p p p
exponents in the numerator.
21. 5 1z 22z3 52z 5 5z1> 2 2z3> 2 5z5> 2 is not a polynomial since the exponents are not integers.
22. 13x 2 4x 52 12x 2 3x 22 13x 2 2x 2 2 1 4x 3x2 15 22 x 2 x 3
23. 14m3 3m2 52 13m3 m2 52 14m3 3m3 2 1 3m2 m2 2 15 52 m3 4m2 10
,756 CHAPTER R Basic Algebraic Concepts
24. 112y 2 8y 62 13y 2 4y 22 12y 2 8y 6 3y 2 4y 2 9y 2 4y 4
25. 18p2 5p2 13p2 2p 42 8p2 5p 3p2 2p 4 5p2 3p 4
26. 16m4 3m2 m2 12m3 5m2 4m2 1m2 m2 6m4 3m2 m 2m3 5m2 4m m2 m
6m4 2m3 7m2 4m
27. 18x 3 x 32 12x 3 x 2 2 14x 2 3x 12 8x 3 x 3 2x 3 x 2 4x 2 3x 1
6x 3 3x 2 4x 4
28. 14r 1217r 22 4r 7r 4r 2 1 7r 1 2 28r 2 8r 7r 2 28r 2 r 2
29. 15m 6213m 42 5m 3m 5m 4 6 3m 6 4 15m 2 20m 18m 24 15m 2 2m 24
30. a3x b a5x b 3x 5x 3x 5x 15x 2 x x 15x 2 x
2 1 1 2 2 1 10 2 7 2
3 3 3 3 3 3 3 9 3 9
31. a2m b a3m b 2m 3m 2m 3m 6m 2 m m 6m 2 m
1 1 1 1 1 1 3 1 1 1
4 2 2 4 4 2 4 8 4 8
32. 4x 2 13x 3 2x 2 5x 12 4x 2 3x 3 4x 2 2x 2 4x 2 15x2 4x 2 1 12x 5 8x 4 20x 3 4x 2
33. 2b3 1b2 4b 32 2b3 b2 2b3 1 4b2 2b3 3 2b5 8b4 6b3
34. 12z 121 z 2 3z 42 2z1 z 2 2 2z13z2 2z1 42 11 z 2 2 113z2 11 42
2z 3 6z 2 8z z 2 3z 4 2z 3 7z 2 11z 4
35. 1m n k21m 2n 3k2 m2 2mn 3km mn 2n2 3kn km 2kn 3k 2
m2 mn 2km 2n2 5kn 3k 2
36. 1r 3s t212r s t2 2r 2 rs r t 6rs 3s2 3st 2r t st t 2
2r 2 7rs 3r t 3s2 4st t 2
37. To find the square of a binomial, find the sum of the square of the first term, twice the product of the two terms,
and the square of the last term.
38. To find the product of the sum and difference of two terms, find the difference between the square of the first
term and the square of the last term.
39. 12m 3212m 32 12m2 2 132 2 4m2 9
40. 18s 3t218s 3t2 18s2 2 13t2 2 64s2 9t 2
41. 14m 2n2 2 14m2 2 214m212n2 12n2 2 16m2 16mn 4n2
42. 1a 6b2 2 1a2 2 21a216b2 16b2 2 a2 12ab 36b2
43. 15r 3t 2 2 2 15r2 2 215r213t 2 2 13t 2 2 2 25r 2 30r t 2 9t 4
44. 12z 4 3y2 2 12z4 2 2 212z4 213y2 13y2 2 4z 8 12z4y 9y 2
45. 3 12p 32 q4 2 12p 32 2 212p 321q2 1q2 2 3 12p2 2 212p2132 32 4 4pq 6q q 2
4p2 12p 9 4pq 6q q 2
46. 3 14y 12 z4 2 14y 12 2 214y 121z2 1z2 2 3 14y2 2 214y2112 12 4 8yz 2z z 2
16y 2 8y 1 8yz 2z z 2
47. 3 13q 52 p4 3 13q 52 p4 13q 52 2 1p2 2 3 13q2 2 213q2152 152 2 4 p2
9q 2 30q 25 p2
, Factoring SECTION R.2 757
48. 3 19r s2 2 4 3 19r s2 24 19r s2 2 122 2 3 19r2 2 219r21s2 1s2 2 4 4 81r 2 18rs s2 4
49. 3 13a b2 14 2 13a b2 2 213a b2112 112 2 3 13a2 2 213a21b2 1b2 2 4 6a 2b 1
9a2 6ab b2 6a 2b 1
50. 3 12m 72 n4 2 12m 72 2 212m 721n2 1n2 2 3 12m2 2 212m2172 172 2 4 4mn 14n n2
4m2 28m 49 4mn 14n n2
51. 16p 5q213p 7q2 16p213p2 16p217q2 15q213p2 15q217q2 18p2 42pq 15pq 35q2
18p2 27pq 35q2
52. 12p 1213p2 4p 52 12p213p2 2 12p21 4p2 12p2152 113p2 2 11 4p2 1152
6p3 8p2 10p 3p2 4p 5 6p3 11p2 14p 5
53. 1p3 4p2 p2 13p2 2p 72 p3 4p2 p 3p2 2p 7 p3 7p2 p 7
54. 16k 32 2 16k2 2 216k2132 132 2 36k 2 36k 9
55. y14x 3y214x 3y2 y3 14x2 2 13y2 2 4 y316x 2 9y 2 4 16x 2y 9y 3
56. 1r 5 r 3 r2 13r 5 4r 4 r 3 2r2 4r 5 4r 4 3r
57. 12z y213z 4y2 12z213z2 12z21 4y2 1y213z2 1y21 4y2 6z 2 8yz 3yz 4y 2
6x 2 5yz 4y 2
58. 17m 2n217m 2n2 17m2 2 12n2 2 49m2 4n2
59. 13p 52 2 13p2 2 213p2152 52 9p2 30p 25
60. 213r 2 4r 22 31 r 2 4r 52 6r 2 8r 4 3r 2 12r 15 9r 2 4r 19
61. p14p 62 213p 82 4p2 6p 6p 16 4p2 16
62. m15m 22 915 m2 5m2 2m 45 9m 5m2 11m 45
63. y1y 2 42 6y 2 12y 32 y 3 4y 12y 3 18y 2 11y 3 18y 2 4y
64. z3 19 z2 4z12 3z2 9z3 z4 8z 12z2 z4 9z 3 12z2 8z
R.2: Factoring
1. (a) 1x 5y2 2 x 2 10xy 25y 2; B
(b) 1x 5y2 2 x 2 10xy 25y 2; C
(c) 1x 5y21x 5y2 x 2 25y 2; A
(d) 15y x215y x2 25y 2 x 2; D
2. (a) 12x 3214x 2 6x 92 8x 3 27; B
(b) 12x 3214x 2 6x 92 8x 3 27; C
(c) 13 2x219 6x 4x 2 2 27 8x 3; A
3. 4k 2m3 8k4m3 12k 2m4 4k 2m3 11 2k 2 3m2
4. 28r4s2 7r 3s 35r 4s3 7r 3s14rs 1 5rs2 2
, 758 CHAPTER R Basic Algebraic Concepts
5. 21a b2 4m1a b2 1a b212 4m2 21a b211 2m2
6. 41y 22 2 31y 22 1y 22141y 22 32 1y 2214y 52
7. 12y 321y 22 1y 521y 22 1y 2212y 3 y 52 1y 2213y 22
8. 16a 121a 22 16a 1213a 12 16a 121a 2 3a 12 16a 1214a 12
9. 15r 621r 32 12r 121r 32 1r 3215r 6 12r 122 1r 3213r 52
10. 13z 221z 42 1z 621z 42 1z 4213z 2 1z 622 1z 4212z 42 21z 421z 22
11. 21m 12 31m 12 2 21m 12 3 1m 1212 31m 12 21m 12 2 2
1m 1212 3m 3 21m2 2m 122 1m 1215 3m 2m2 4m 22 1m 1212m2 7m 72
12. 51a 32 3 21a 32 1a 32 2 1a 32151a 32 2 2 1a 322
1a 32151a2 6a 92 1 a2 1a 3215a2 30a 45 1 a2 1a 3215a2 31a 462
13. 6st 9t 10s 15 3t12s 32 512s 32 12s 3213t 52
14. 10ab 6b 35a 21 2b15a 32 715a 32 15a 3212b 72
15. 10x 2 12y 15x 8xy 10x 2 15x 8xy 12y 5x12x 32 4y12x 32 12x 3215x 4y2
16. 2m4 6 am4 3a 21m4 32 a1m4 32 1m4 3212 a2
17. t 3 2t 2 3t 6 t 2 1t 22 31t 22 1t 221t 2 32
18. x 3 3x 2 5x 15 x 2 1x 32 51x 32 1x 321x 2 52
19. 18a 3212a 52 311 8a 32 4 3 112a 52 4 13 8a215 2a2; Both are correct.
20. 6a2 48a 120 61a2 8a 202 61a 1021a 22
21. 8h2 24h 320 81h2 3h 402 81h 821h 52
22. 3m3 12m2 9m 3m1m2 4m 32 3m1m 321m 12
23. 9y 4 54y 3 45y 2 9y 2 1y 2 6y 52 9y 2 1y 121y 52
24. 6k 2 5kp 6p2 13k 2p212k 3p2
25. 14m2 11mr 15r 2 17m 5r212m 3r2
26. 5a2 7ab 6b2 15a 3b21a 2b2
27. 12s2 11st 5t 2 13s t214s 5t2
28. 9x 2 6x 3 x 4 x 2 19 6x x 2 2 x 2 13 x2 2
29. 30a2 am m2 15a m216a m2
30. 24a4 10a3b 4a2b2 2a2 112a2 5ab 2b2 2 2a2 13a 2b214a b2
31. 18x 5 15x 4z 75x 3z 2 3x 3 16x 2 5xz 25z 2 2 3x 3 12x 5z213x 5z2
32. 9m2 12m 4 13m 22 2
33. 16p2 40p 25 14p 52 2
34. 32a2 48ab 18b2 2116a2 24ab 9b2 2 214a 3b2 2
35. 20p2 100pq 125q 2 514p2 20pq 25q2 2 512p 5q2 2
36. 4x 2y 2 28xy 49 12xy 72 2
37. 9m2n2 12mn 4 13mn 22 2
R.1: Exponents and Polynomials
1. 1 42 3 1 42 2 1 42 32 1 42 5
2. 152 2 152 6 152 26 152 8
3. 20 1
4. 20 112 1
5. 15m2 0 1, if m 0
6. 1 4z2 0 1, if z 0
7. 122 2 5 225 210
8. 164 2 3 643 612
9. 12x 5y 4 2 3 23 1x 5 2 3 1y 4 2 3 23x 15y 12 or 8x 15y 12
10. 1 4m3n9 2 2 1 42 2 1m3 2 2 1n9 2 2 42m6n18 or 16m6n18
p4 2 p42 p8
11. a b a 2 b 2
q q q
r 8 3 r 83 r 24
12. a 2
b 23 6
s s s
13. 5x 11 is a polynomial. It is a monomial since it has one term. It has degree 11 since 11 is the highest exponent.
14. 9y 12 y 2 is a polynomial. It is a binomial since it has two terms. It has degree 12 since 12 is the highest exponent.
15. 18p5q 6pq is a polynomial. It is a binomial since it has two terms. It has degree 6 since 6 is the sum of the
exponents in the term 18p5q. (The term 6pq has degree 2.)
16. 2a6 5a2 4a is a polynomial. It is a trinomial since it has three terms. It has degree 6 since 6 is the highest
exponent.
17. 12 x 2 13 x 6 is a polynomial. It is a binomial since it has two terms. It has degree 6 since 6 is the highest
exponent.
18. 17 m5n2 213 m3n2 is a polynomial. It is a binomial since it has two terms. It has degree 7 since 7 is the
sum of the exponents in the term 17 m5n2. (The other term has degree 5.)
1 22 3 42
19. r s r s rs3 is a polynomial. It is a trinomial since it has three terms. It has degree 6 since 6 is the
3 5
3
sum of the exponents in the term r 4s 2. (The other terms have degree 4.)
5
5 2 5
20. 2 3 is not a polynomial since positive exponents in the denominator are equivalent to negative
p p p
exponents in the numerator.
21. 5 1z 22z3 52z 5 5z1> 2 2z3> 2 5z5> 2 is not a polynomial since the exponents are not integers.
22. 13x 2 4x 52 12x 2 3x 22 13x 2 2x 2 2 1 4x 3x2 15 22 x 2 x 3
23. 14m3 3m2 52 13m3 m2 52 14m3 3m3 2 1 3m2 m2 2 15 52 m3 4m2 10
,756 CHAPTER R Basic Algebraic Concepts
24. 112y 2 8y 62 13y 2 4y 22 12y 2 8y 6 3y 2 4y 2 9y 2 4y 4
25. 18p2 5p2 13p2 2p 42 8p2 5p 3p2 2p 4 5p2 3p 4
26. 16m4 3m2 m2 12m3 5m2 4m2 1m2 m2 6m4 3m2 m 2m3 5m2 4m m2 m
6m4 2m3 7m2 4m
27. 18x 3 x 32 12x 3 x 2 2 14x 2 3x 12 8x 3 x 3 2x 3 x 2 4x 2 3x 1
6x 3 3x 2 4x 4
28. 14r 1217r 22 4r 7r 4r 2 1 7r 1 2 28r 2 8r 7r 2 28r 2 r 2
29. 15m 6213m 42 5m 3m 5m 4 6 3m 6 4 15m 2 20m 18m 24 15m 2 2m 24
30. a3x b a5x b 3x 5x 3x 5x 15x 2 x x 15x 2 x
2 1 1 2 2 1 10 2 7 2
3 3 3 3 3 3 3 9 3 9
31. a2m b a3m b 2m 3m 2m 3m 6m 2 m m 6m 2 m
1 1 1 1 1 1 3 1 1 1
4 2 2 4 4 2 4 8 4 8
32. 4x 2 13x 3 2x 2 5x 12 4x 2 3x 3 4x 2 2x 2 4x 2 15x2 4x 2 1 12x 5 8x 4 20x 3 4x 2
33. 2b3 1b2 4b 32 2b3 b2 2b3 1 4b2 2b3 3 2b5 8b4 6b3
34. 12z 121 z 2 3z 42 2z1 z 2 2 2z13z2 2z1 42 11 z 2 2 113z2 11 42
2z 3 6z 2 8z z 2 3z 4 2z 3 7z 2 11z 4
35. 1m n k21m 2n 3k2 m2 2mn 3km mn 2n2 3kn km 2kn 3k 2
m2 mn 2km 2n2 5kn 3k 2
36. 1r 3s t212r s t2 2r 2 rs r t 6rs 3s2 3st 2r t st t 2
2r 2 7rs 3r t 3s2 4st t 2
37. To find the square of a binomial, find the sum of the square of the first term, twice the product of the two terms,
and the square of the last term.
38. To find the product of the sum and difference of two terms, find the difference between the square of the first
term and the square of the last term.
39. 12m 3212m 32 12m2 2 132 2 4m2 9
40. 18s 3t218s 3t2 18s2 2 13t2 2 64s2 9t 2
41. 14m 2n2 2 14m2 2 214m212n2 12n2 2 16m2 16mn 4n2
42. 1a 6b2 2 1a2 2 21a216b2 16b2 2 a2 12ab 36b2
43. 15r 3t 2 2 2 15r2 2 215r213t 2 2 13t 2 2 2 25r 2 30r t 2 9t 4
44. 12z 4 3y2 2 12z4 2 2 212z4 213y2 13y2 2 4z 8 12z4y 9y 2
45. 3 12p 32 q4 2 12p 32 2 212p 321q2 1q2 2 3 12p2 2 212p2132 32 4 4pq 6q q 2
4p2 12p 9 4pq 6q q 2
46. 3 14y 12 z4 2 14y 12 2 214y 121z2 1z2 2 3 14y2 2 214y2112 12 4 8yz 2z z 2
16y 2 8y 1 8yz 2z z 2
47. 3 13q 52 p4 3 13q 52 p4 13q 52 2 1p2 2 3 13q2 2 213q2152 152 2 4 p2
9q 2 30q 25 p2
, Factoring SECTION R.2 757
48. 3 19r s2 2 4 3 19r s2 24 19r s2 2 122 2 3 19r2 2 219r21s2 1s2 2 4 4 81r 2 18rs s2 4
49. 3 13a b2 14 2 13a b2 2 213a b2112 112 2 3 13a2 2 213a21b2 1b2 2 4 6a 2b 1
9a2 6ab b2 6a 2b 1
50. 3 12m 72 n4 2 12m 72 2 212m 721n2 1n2 2 3 12m2 2 212m2172 172 2 4 4mn 14n n2
4m2 28m 49 4mn 14n n2
51. 16p 5q213p 7q2 16p213p2 16p217q2 15q213p2 15q217q2 18p2 42pq 15pq 35q2
18p2 27pq 35q2
52. 12p 1213p2 4p 52 12p213p2 2 12p21 4p2 12p2152 113p2 2 11 4p2 1152
6p3 8p2 10p 3p2 4p 5 6p3 11p2 14p 5
53. 1p3 4p2 p2 13p2 2p 72 p3 4p2 p 3p2 2p 7 p3 7p2 p 7
54. 16k 32 2 16k2 2 216k2132 132 2 36k 2 36k 9
55. y14x 3y214x 3y2 y3 14x2 2 13y2 2 4 y316x 2 9y 2 4 16x 2y 9y 3
56. 1r 5 r 3 r2 13r 5 4r 4 r 3 2r2 4r 5 4r 4 3r
57. 12z y213z 4y2 12z213z2 12z21 4y2 1y213z2 1y21 4y2 6z 2 8yz 3yz 4y 2
6x 2 5yz 4y 2
58. 17m 2n217m 2n2 17m2 2 12n2 2 49m2 4n2
59. 13p 52 2 13p2 2 213p2152 52 9p2 30p 25
60. 213r 2 4r 22 31 r 2 4r 52 6r 2 8r 4 3r 2 12r 15 9r 2 4r 19
61. p14p 62 213p 82 4p2 6p 6p 16 4p2 16
62. m15m 22 915 m2 5m2 2m 45 9m 5m2 11m 45
63. y1y 2 42 6y 2 12y 32 y 3 4y 12y 3 18y 2 11y 3 18y 2 4y
64. z3 19 z2 4z12 3z2 9z3 z4 8z 12z2 z4 9z 3 12z2 8z
R.2: Factoring
1. (a) 1x 5y2 2 x 2 10xy 25y 2; B
(b) 1x 5y2 2 x 2 10xy 25y 2; C
(c) 1x 5y21x 5y2 x 2 25y 2; A
(d) 15y x215y x2 25y 2 x 2; D
2. (a) 12x 3214x 2 6x 92 8x 3 27; B
(b) 12x 3214x 2 6x 92 8x 3 27; C
(c) 13 2x219 6x 4x 2 2 27 8x 3; A
3. 4k 2m3 8k4m3 12k 2m4 4k 2m3 11 2k 2 3m2
4. 28r4s2 7r 3s 35r 4s3 7r 3s14rs 1 5rs2 2
, 758 CHAPTER R Basic Algebraic Concepts
5. 21a b2 4m1a b2 1a b212 4m2 21a b211 2m2
6. 41y 22 2 31y 22 1y 22141y 22 32 1y 2214y 52
7. 12y 321y 22 1y 521y 22 1y 2212y 3 y 52 1y 2213y 22
8. 16a 121a 22 16a 1213a 12 16a 121a 2 3a 12 16a 1214a 12
9. 15r 621r 32 12r 121r 32 1r 3215r 6 12r 122 1r 3213r 52
10. 13z 221z 42 1z 621z 42 1z 4213z 2 1z 622 1z 4212z 42 21z 421z 22
11. 21m 12 31m 12 2 21m 12 3 1m 1212 31m 12 21m 12 2 2
1m 1212 3m 3 21m2 2m 122 1m 1215 3m 2m2 4m 22 1m 1212m2 7m 72
12. 51a 32 3 21a 32 1a 32 2 1a 32151a 32 2 2 1a 322
1a 32151a2 6a 92 1 a2 1a 3215a2 30a 45 1 a2 1a 3215a2 31a 462
13. 6st 9t 10s 15 3t12s 32 512s 32 12s 3213t 52
14. 10ab 6b 35a 21 2b15a 32 715a 32 15a 3212b 72
15. 10x 2 12y 15x 8xy 10x 2 15x 8xy 12y 5x12x 32 4y12x 32 12x 3215x 4y2
16. 2m4 6 am4 3a 21m4 32 a1m4 32 1m4 3212 a2
17. t 3 2t 2 3t 6 t 2 1t 22 31t 22 1t 221t 2 32
18. x 3 3x 2 5x 15 x 2 1x 32 51x 32 1x 321x 2 52
19. 18a 3212a 52 311 8a 32 4 3 112a 52 4 13 8a215 2a2; Both are correct.
20. 6a2 48a 120 61a2 8a 202 61a 1021a 22
21. 8h2 24h 320 81h2 3h 402 81h 821h 52
22. 3m3 12m2 9m 3m1m2 4m 32 3m1m 321m 12
23. 9y 4 54y 3 45y 2 9y 2 1y 2 6y 52 9y 2 1y 121y 52
24. 6k 2 5kp 6p2 13k 2p212k 3p2
25. 14m2 11mr 15r 2 17m 5r212m 3r2
26. 5a2 7ab 6b2 15a 3b21a 2b2
27. 12s2 11st 5t 2 13s t214s 5t2
28. 9x 2 6x 3 x 4 x 2 19 6x x 2 2 x 2 13 x2 2
29. 30a2 am m2 15a m216a m2
30. 24a4 10a3b 4a2b2 2a2 112a2 5ab 2b2 2 2a2 13a 2b214a b2
31. 18x 5 15x 4z 75x 3z 2 3x 3 16x 2 5xz 25z 2 2 3x 3 12x 5z213x 5z2
32. 9m2 12m 4 13m 22 2
33. 16p2 40p 25 14p 52 2
34. 32a2 48ab 18b2 2116a2 24ab 9b2 2 214a 3b2 2
35. 20p2 100pq 125q 2 514p2 20pq 25q2 2 512p 5q2 2
36. 4x 2y 2 28xy 49 12xy 72 2
37. 9m2n2 12mn 4 13mn 22 2
