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Alg II - Rationals ~1~ NJCTL.org
PMI Rational Expressions & Equations Unit
Variation
Class Work
1. y varies inversely with x. If 𝑦 = 12 when 𝑥 = 4, find y when 𝑥 = −6.
2. y varies inversely with x. If 𝑦 = 8 when 𝑥 = 3, find x when𝑦 = −2.
3. y varies inversely with x2. If 𝑦 = 3 when𝑥 = 5, find y when𝑥 = 3.
4. Circumference varies directly with area of a circle and inversely with the radius. Find the constant of variation if 𝐶 = 18
when 𝐴 = 27 and 𝑟 = 3. What is radius when 𝐶 = 25 and 𝐴 = 50?
5. y varies jointly with x and z. If 𝑦 = 24 when 𝑥 = 4 and 𝑧 = 2, find y when 𝑥 = −6 and 𝑧 = 2.
6. y varies jointly with x and z. If 𝑦 = 32 when 𝑥 = 4 and 𝑧 = 2, find x when 𝑦 = 48 and 𝑧 = 2.
7. V varies jointly with r2 and h. If 𝑉 = 24𝜋 when ℎ = 6 and 𝑟 = 2, find r when 𝑉 = 18𝜋 and ℎ = 2.
Variation
Homework
8. y varies inversely with x. If 𝑦 = 9 when𝑥 = 4, find y when𝑥 = −6.
9. y varies inversely with x. If 𝑦 = 8 when 𝑥 = 5, find x when 𝑦 = −2.
10. y varies inversely with x2. If 𝑦 = 3, when 𝑥 = 4, find y when 𝑥 = 3.
11. Area of a triangle varies jointly with its height and base. Find the constant of variation if 𝐴 = 16 when ℎ = 4 and 𝑏 = 8.
What is base when 𝐴 = 9 and ℎ = 3?
12. y varies jointly with x and z. If 𝑦 = 24 when 𝑥 = 6 and 𝑧 = 2, find y when 𝑥 = −6 and 𝑧 = 3.
13. y varies jointly with x and z. If 𝑦 = 40 when 𝑥 = −4 and 𝑧 = 2, find x when𝑦 = 60 and 𝑧 = 2.
14. V varies jointly with r2 and h. 𝑉 = 36𝜋 when ℎ = 4 and 𝑟 = 3, find r when𝑉 = 80𝜋 and ℎ = 5.
Alg II - Rationals ~2~ NJCTL.org
Reducing Rational Expressions
Class Work
Simplify.
15. 3x
6 16.
40b
12b 17.
12x2
4x 18.
18a3b2
14a5b6
19. 60j4k6m8
16j3k6m9 20. 12c2− 6
3 21.
8n2+ 4n
6n2 + 3n 22.
5h − 10
4h − 8
23. . v2 − 4v + 4
v2 − 4 24.
f2 + 7f + 12
f2 − 2f − 15 25.
4s3 − 20s2 + 24s
16 − 8s 26.
2d2 − 7d + 6
3d2 − 8d + 4
27. 4𝑥3 − 4𝑥2 − 15𝑥
2𝑥5 − 3𝑥4 − 5𝑥3 28. 54𝑥4 − 6𝑥2
54𝑥3 − 72𝑥2 + 18𝑥 29.
4𝑚3 − 4
2𝑚2 + 2𝑚 + 2 30.
3𝑝2 + 7𝑝 − 6
𝑝2 + 𝑝 − 6
31. 8𝑎2 + 4𝑎 − 60
4𝑎 − 10 32.
2𝑥3 + 2𝑦3
4𝑥2 − 4𝑦2 33. 12𝑎2𝑏2 − 12
4𝑎𝑏𝑐 − 4𝑐 34.
15𝑥3 + 7𝑥2 − 2𝑥
3𝑥4 + 2𝑥3
35. 3𝑝2 − 𝑝 − 2
4 − 4𝑝
36. 6𝑥3 + 15𝑥2+ 9𝑥
12𝑥 + 6𝑥2 − 6𝑥3
Alg II - Rationals ~3~ NJCTL.org
Reducing Rational Expressions
Homework
Simplify.
37. 9y
6 38.
48c
16c 39.
12x2
8x3 40.
28a3b2
14a4b8
41. 60j4k6m8
12j2k6m10 42.
12c2− 9
6 43.
10n2− 5n
15n2 − 35n 44.
12h − 8
9h − 6
45. v2 − 4v + 3
v2 − 9 46.
f2+ 12f + 27
f2 + f − 72 47.
5rs3 − 20rs2 + 15rs
15r − 5rs 48.
4d2 + 5d + 1
3d2 + 5d + 2
49. 𝑥4− 𝑥2
𝑥4 − 𝑥3 50. 2𝑎3 − 2𝑏3
12𝑎𝑐 − 12𝑏𝑐 51.
4𝑚2 − 4𝑚𝑛 + 4𝑛2
𝑚3 + 𝑛3 52. 8𝑚𝑝2− 2𝑚
4𝑝2 − 1
53. 12𝑥3 − 10𝑥2 + 2𝑥
2𝑥5 + 𝑥4 − 𝑥3 54.
6𝑘2𝑙2 − 5𝑘𝑙3 + 𝑙4
4𝑘2 − 𝑙2 55.
𝑥2𝑦 − 4𝑥𝑦
𝑥4𝑦3 − 2𝑥3𝑦3 56.
12𝑘3− 4𝑠𝑘3
4𝑘𝑠2 − 6𝑘𝑠 − 18𝑘
57. 6𝑝2− 15𝑝 + 6
18 − 30𝑝 − 12𝑝2 58. −𝑥 + 2
4𝑥2 − 7𝑥 − 2
Alg II - Rationals ~4~ NJCTL.org
Multiplying & Dividing Rational Expressions
Class Work
Perform the indicated operation. Write answer in simplified form.
59. 8𝑎 ∙11
12𝑎 60.
12𝑏
5𝑐∙
15𝑐2
8𝑏3 61.
𝑑 + 𝑒
𝑓 + 2∙
𝑓2 − 4
(𝑑 + 𝑒)2
62. 𝑔2− 7𝑔 + 12
𝑔2− 9∙
𝑔2+ 4𝑔 + 4
2𝑔2 + 6𝑔 + 4 63.
g + 5
h + 3∙
h2 − h − 12
g2 − 25∙
g2 − 3g − 10
h2− 5h + 4 64.
14h
15÷ 6h
65. 5𝑗2 ÷10𝑗
9 66.
𝑘 + 𝑙
𝑘2+ 2𝑘𝑙 + 𝑙2 ÷2𝑘 + 3𝑙
𝑘2 − 𝑙2 67. 𝑚 + 5
𝑚2 + 7𝑚 + 10÷
𝑚2 + 5𝑚 + 6
𝑚2 − 4
68.
2
n2 − 44
n − 2
69. p2 + 4p + 3
p2 + 7p + 10÷
p2− 1
p2+ 2p + 1∙
p − 1
p2 + 5p + 6
70. 4q2r
12q5r3 ÷16q5r4
18q3r8 ÷8q6r3
24qr∙
q2r5
q5r2 ∙ 2
71. 𝑥3 − 27
𝑥2 − 9÷
4𝑥 − 2
2𝑥2 + 5𝑥 − 3 72.
3 − 𝑚
2𝑚 + 1∙
6𝑚2+ 𝑚 − 1
𝑚2− 9 73.
𝑚2 − 𝑛2
3𝑚2 − 5𝑚𝑛 + 2𝑛2 ∙3𝑚𝑛2 − 2𝑛3
𝑛
74. 4𝑎2− 4𝑎𝑏 + 4𝑏2
𝑎3+ 𝑏3 ∙𝑎2− 𝑏2
12𝑎 − 12𝑏 75.
6𝑚2+ 9𝑚 + 3
6𝑚2+ 24𝑚 + 18 ÷
2𝑚2+ 7𝑚 + 3
2𝑚2+ 3𝑚 − 9 76.
8 + 2𝑥 − 𝑥2
𝑥2 − 𝑥 − 12∙
𝑥 − 2
3𝑥2 + 5𝑥 − 2
77. A rectangular shaped “dartboard” has dimensions (3x +3) by (2x+1) inches. On the board is a square with sides (x+1)
inches. What is the probability that a randomly thrown dart that hits board lands in the square?
Alg II - Rationals ~5~ NJCTL.org
Multiplying & Dividing Rational Expressions
Homework
Perform the indicated operation. Write answer in simplified form.
78. 9𝑎 ∙11
12𝑎 79.
18𝑏
5𝑐∙
20𝑐5
8𝑏4 80.
𝑑 + 𝑒
𝑓 − 2∙
𝑓2 − 4
(𝑑 + 𝑒)3
81. 𝑔2− 8𝑔 + 15
𝑔2− 25∙
𝑔2+ 10𝑔 + 25
2𝑔2− 4𝑔 − 6 82.
g + 4
h + 7∙
h2+ 3h − 28
g2− 16∙
g2+ 10g + 21
h2− 7h + 12 83.
10h
15÷ 8h
84. 6𝑗2 ÷12𝑗
9 85.
2𝑘 + 3𝑙
𝑘2− 2𝑘𝑙 + 𝑙2 ÷2𝑘 + 3𝑙
𝑘2 − 𝑙2 86. 𝑚 + 6
𝑚2+ 7𝑚 +12÷
𝑚2+ 5𝑚 − 6
𝑚2− 9
87.
n2− 1
n2− 4n + 4n − 1
n2− 4
88.
p2+ 4p + 4
p2+ 7p + 12÷
p2− 4
p2+ 5p + 4∙
p− 2
p2+ 4p + 3 89.
10q9r
15q2r7 ÷20q7r12
16q2r10 ÷8q7r5
25q3r∙
q3r4
q8r3 ∙ 𝑟
90. 27𝑥3− 𝑦3
18𝑥2+ 6𝑥𝑦 + 2𝑦2 ∙4𝑥2− 𝑦2
6𝑥2− 5𝑥𝑦 + 𝑦2 91. 𝑚𝑛3+ 𝑛4
3𝑚𝑛4− 𝑛5 ÷3𝑚2− 𝑚𝑛 − 4𝑛2
6𝑚2− 11𝑚𝑛 + 4𝑛2 92. 8𝑎𝑐 − 2𝑏𝑐
𝑎2− 𝑏2 ÷20𝑎𝑐2− 5𝑏𝑐2
25𝑎 − 25𝑏
93. 6𝑥2− 17𝑥 + 5
4𝑥2− 9𝑥 + 2÷
6𝑥2+ 7𝑥 − 3
3𝑥2− 𝑥 − 10 94.
3 − 𝑥
𝑥2+ 𝑥 − 20∙
2𝑥2− 7𝑥 − 4
2𝑥2− 5𝑥 − 3 95.
8 − 2𝑚 − 3𝑚2
5𝑚2− 3𝑚3 ÷4 − 3𝑚
3𝑚5 + 𝑚4− 10𝑚3
96. A rectangular shaped “dartboard” has dimensions (4x+6) by (2x+3) inches. On the board is a square with sides
(2x+3) inches. What is the probability that a randomly thrown dart that hits board lands in the square?
Alg II - Rationals ~6~ NJCTL.org
Adding and Subtracting Rational Expressions
Class Work
Perform the indicated operation. Write answer in simplified form.
97. 5
2x+
3
2x 98.
6y
4−
2y
4 99.
z − 3
6+
2z
6
100. 4w + 7
2w + 1−
2w + 6
2w + 1 101.
5v + 1
2v + 3+
v + 8
2v + 3 102.
3(u + 2)
2u − 1−
5(u + 1)
2u − 1
103. 3
x + 4+
2
x − 4 104.
5
x2− 9−
2x
x − 3 105.
5
x2 + 4x + 4+
6
x2− 4
106. 2
x − 3+
3
x2− 7x + 12+
4
x − 4 107.
𝑥2 − 2
3𝑥2 − 𝑥−
𝑥2 − 3
3𝑥 − 1
108. 3𝑥 − 𝑥 − 4
𝑥2 + 4𝑦
109. 5
x2− 5x + 6+
4
x2+ 3x − 10−
3
𝑥2+ 2𝑥 − 15 110.
2𝑚 + 4
𝑚 − 6+
3𝑚 − 10
6 − 𝑚 111.
2𝑥2 − 𝑥
4 − 𝑥−
2𝑥2
𝑥 − 4−
𝑥2
𝑥
112. 4𝑥 − 2
2𝑥2 − 5𝑥 + 3−
3𝑥 − 5
2𝑥2 + 𝑥 − 6+
𝑥2
𝑥2 + 𝑥 − 2 113.
4𝑦 − 9
3𝑦 + 2− 5𝑦
Alg II - Rationals ~7~ NJCTL.org
Adding and Subtracting Rational Expressions
Homework
Perform the indicated operation. Write answer in simplified form.
114. 4
3x+
8
3x 115.
7y
5−
2y
5 116.
5z − 4
6+
5z
6
117. 5w + 4
3w + 2−
2w + 2
3w + 2 118.
3v + 7
4v + 6+
v + 9
4v + 6 119.
4(2u + 1)
2u − 1−
2(u + 8)
2u − 1
120. 3
3t − 1+
2
2t + 2 121.
5
x2− 5x + 6−
2x
x − 3 122.
5
x2+ 6x + 9+
6
x2− 9
123. 2
x + 3+
3
x2− 3x − 18+
4
x − 6 124.
5
x2− 4x + 3+
4
x2+ x − 12−
3
𝑥2+ 3𝑥 − 4 125. 4 −
2𝑥 − 3
3𝑥 − 2
126. 4𝑥
𝑥 − 3+
6𝑥 − 5
2𝑥2 − 6𝑥 127.
6𝑝 − 2
4 − 𝑝−
𝑝 − 1
𝑝 − 4 128.
8𝑏 + 4
3 − 2𝑏−
6𝑏 + 2
2𝑏 − 3
129. 3𝑚 − 2
16𝑚2 − 1−
6𝑚 + 5
8𝑚2 − 6𝑚 + 1 130.
2𝑦 − 7
6𝑦2 + 𝑦 −1−
𝑦2 − 𝑦
2𝑦2 − 𝑦 − 1
Alg II - Rationals ~8~ NJCTL.org
Solving Rational Equations
Class Work
Solve for x. Check for extraneous solutions.
131. 2
x + 3=
3
x − 2 132.
4
2x − 1=
6
x + 5
133. 2x − 1
2+
x + 3
10=
6x
5 134.
5
2x−
x + 3
x=
3
4
135. 2
x + 3+
5
2=
1
x + 3 136.
2
x − 3+
4x
x2− 9=
−1
x + 3
137. 3
x + 2−
4
x − 1=
5
x2+ x − 2 138.
x
x + 5−
2
x − 3=
1
x2+ 2x − 15
139. 2
x2− 4+
1
x − 2=
3
x + 2−
5
x2− 4x + 4 140.
30
5𝑥+ 3 =
6
𝑥
141. 5
2𝑥+
7
3=
2
3𝑥−
4
6 142.
4
3𝑥 − 2+
2
𝑥 − 1=
12
3𝑥2 − 5𝑥 + 2
143. 3
𝑥3+ 27+
2
𝑥 + 3=
5
𝑥2 − 3𝑥 + 9
Alg II - Rationals ~9~ NJCTL.org
Solving Rational Equations
Homework
Solve for x. Check for extraneous solutions.
144. 2
x − 1=
5
x + 4 145.
5
3x + 4=
6
3x + 6
146. 3x + 1
3+
4 − x
2=
5x
6 147.
5
3x−
x + 3
x=
3
2
148. 2
x − 3+
5
3=
1
x − 3 149.
3
x − 2+
2x
x2− 4=
16
x + 2
150. 7
x + 5−
2
x − 2=
3
x2+ 3x − 10 151.
x
2x + 1−
2
x − 1=
x+2
2x2− x − 1
152. 2
x2+ 4x + 3+
1
x + 3=
3
x + 1−
5
x2+ 6x + 9 153.
2𝑥
4+
5
𝑥=
7
𝑥
154. 3
2𝑥 − 1+
4
2𝑥 + 1=
3
4𝑥2− 1 155.
−4
𝑥 − 1+
2
𝑥3 − 1=
8
𝑥2 + 𝑥 + 1
156. 4
𝑥+
3
4𝑥=
−2
3+
5
𝑥
Alg II - Rationals ~10~ NJCTL.org
Graphing Rational Expressions
Class Work
Graph the following functions. Clearly indicate any x-intercepts, y-intercepts, asymptotes and holes (discontinuities) on
the graph, noting them in the spaces provided below.
157. 𝑓(𝑥) =2
𝑥−1
x-intercepts: ____________________ y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
158. 𝑔(𝑥) =−3
𝑥+2
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
159. ℎ(𝑥) =𝑥+1
𝑥2−1
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
160. 𝑓(𝑥) =𝑥−1
(𝑥−1)(𝑥+2)
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
161. 𝑔(𝑥) =𝑥2+5𝑥+6
𝑥2+3𝑥+2
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
162. ℎ(𝑥) =𝑥2−𝑥−6
𝑥2−5𝑥+6
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Alg II - Rationals ~11~ NJCTL.org
Graphing Rational Expressions
Homework
Graph the following functions. Clearly indicate any x-intercepts, y-intercepts, asymptotes and holes (discontinuities) on
the graph, noting them in the spaces provided below.
163. 𝑓(𝑥) =2
𝑥+3
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
164. 𝑔(𝑥) =−3
𝑥−4
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
165. 𝑓(𝑥) =𝑥+2
(𝑥−1)(𝑥+2)
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
166. ℎ(𝑥) =𝑥−2
𝑥2−4
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
167. 𝑔(𝑥) =𝑥2+9𝑥+18
𝑥2+7𝑥+6
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
168. ℎ(𝑥) =𝑥2+5𝑥−14
𝑥2+6𝑥−7
x-intercepts: ____________________
y-intercepts: ____________________
Holes: _________________________
Vertical asymptotes: ______________
Horizontal asymptotes: ____________
Alg II - Rationals ~12~ NJCTL.org
Unit Review - Multiple Choice
1. Simplify 2𝑥2− 10𝑥 + 12
4𝑥2− 12𝑥
a. x − 2
2
b. x − 2
2x
c. (x − 3)(x − 2)
2x2− 6x
d. (x − 6)(x + 1)
2x(x − 3)
2. Simplify x2+ 15x + 56
x2−49∙
x2− 10x + 21
x2 + 11x + 24
a. (x − 7)(x − 3)
(x + 7)(x + 3)
b. (x + 7)(x − 3)
(x − 7)(x + 3)
c. x − 3
x + 3
d. 1
3. Simplify 6𝑚6𝑛3
4𝑚2𝑛9 ÷9𝑚4𝑛2
8𝑚3𝑛7
a. 4m3
3n
b. 4m5
3n11
c. 3n
4m3
d. 3n11
4m5
4. Simplify 2
3x2 −5
6x
a. −1
6x2
b. 4−5x
6x2
c. −1
6x
d. −1
3x2
5. Simplify 2
x2− 16+
4
x2+ 8x + 16
a. 6x−24
(x − 4)(x + 4)(x + 4)
b. 6
(x + 4)(x + 4)
c. 6x−8
(x − 4)(x + 4)(x + 4)
d. 6x
(x − 4)(x + 4)(x + 4)
Alg II - Rationals ~13~ NJCTL.org
6. The function ℎ(𝑥) =4𝑥2− 3𝑥 − 1
4𝑥2 − 1 has which of the following discontinuities?
a. Vertical asymptotes at 𝑥 = ±1
2
b. Removable discontinuity at 𝑥 = ±1
2
c. Vertical asymptote at 𝑥 =1
2; removable discontinuity at 𝑥 = −
1
2
d. Vertical asymptote at 𝑥 = −1
2; removable discontinuity at 𝑥 =
1
2
7. h varies inversely with t. If ℎ = 8 when 𝑡 = 6, find t when ℎ = 16.
a. 2
b. 3
c. 8
3
d. 48
8. The volume of a cone varies jointly to its height and the square of the radius of its base. If V = 18𝜋 when ℎ = 6
and 𝑟 = 3. What is the radius when V = 12𝜋 and h = 4?
a. 1
b. 3
c. 6
d. 9
9. Simplify 𝑥+3
𝑥+5−
4
𝑥−5+
3𝑥−7
𝑥2−25
a. 𝑥2−3𝑥+3
𝑥2−25
b. 𝑥2−2𝑥−15
𝑥2−25
c. 4𝑥−8
𝑥2−25
d. 𝑥2−3𝑥−42
𝑥2−25
10. Solve: 3
x − 2=
4
x + 2
a. 4
b. 8
c. 14
d. no solution
11. Solve: 4x
x2− 1+
4
x − 1=
2
x + 1
a. -1
b. 1
c. 6
d. no solution
12. Solve: 2
x2− 9+
3
x2 − x − 6=
4
x2+ 5x + 6
a. -25
b. -8
c. 8
d. no solution
Alg II - Rationals ~14~ NJCTL.org
Extended Response
1. At math camp the lap pool is a rectangle that is (𝑥2 − 16)𝑓𝑡 by (𝑥 + 3)𝑓𝑡, the wading pool is a square with sides
(𝑥 + 4)𝑓𝑡.
a. How many times larger is the lap pool than the wading pool?
b. If the wading pool is (𝑥 − 4)𝑓𝑡 deep, what is the pool’s volume?
c. If the lap pool has a depth of (𝑥 + 4)𝑓𝑡, how many times larger is the volume of the lap pool to the
wading pool?
2. Determine each of the following for the graph of the rational function and graph the function.
𝑓(𝑥) =𝑥2+𝑥−6
−4𝑥2−16𝑥−12
x-intercepts: ________________________
y-intercepts: ________________________
holes: _____________________________
vertical asymptotes: __________________
horizontal asymptotes: ________________
Alg II - Rationals ~15~ NJCTL.org
PMI Rational Expressions, Equations and Functions- SOLUTIONS:
Variation: Classwork:
1. 𝑘 = 48; 𝑦 = −8 2. 𝑘 = 24; 𝑥 = −12 3. 𝑘 = 75; 𝑦 = 8.3
4. 𝑘 =0.6
54
5. 𝑘 = 3; 𝑦 = −36 6. 𝑘 = 4; 𝑥 = 6 7. 𝑘 = 𝜋; 𝑟 = 3
Variation: Homework:
8. 𝑘 = 36; 𝑦 = −6 9. 𝑘 = 40; 𝑥 = −20 10. 𝑘 = 48; 𝑦 = 5.3 11. 𝑘 = 0.5; 𝑏 = 6 12. 𝑘 = 2; 𝑦 = −36 13. 𝑘 = −5; 𝑥 = −6 14. 𝑘 = 𝜋; 𝑟 = 4
Reducing Rational Expressions: Classwork:
15. 𝑥
2
16. 10
3
17. 3𝑥
18. 9
7𝑎2𝑏4
19. 15𝑗
4𝑚
20. 2(2𝑐2 – 1)
21. 4
3
22. 5
4
23. 𝑣 − 2
𝑣 + 2
24. 𝑓 + 4
𝑓 − 5
25. –𝑠2− 3𝑠
2
26. 2𝑑 − 3
3𝑑 − 2
27. 2𝑥 + 3
𝑥2(𝑥 + 1)
28. 𝑥(3𝑥 + 1)
3(𝑥 −1)
29. 2(𝑚 − 1)
30. 3𝑝 − 2
𝑝 − 2
31. 2(𝑎 + 3)
32. 𝑥2− 𝑥𝑦 + 𝑦2
2(𝑥 − 𝑦)
33. 3(𝑎𝑏 + 1)
𝑐
34. 5𝑥 − 1
𝑥2
35. −3𝑝 + 2
4
36. −2𝑥 + 3
2(𝑥 − 2)
Reducing Rational Expressions: Homework:
37. 3𝑦
2
38. 3
39. 3
2𝑥
40. 2
𝑎𝑏6
41. 5𝑗2
𝑚2
42. 4𝑐2−3
2
43. 2𝑛−1
3𝑛−7
44. 4
3
45. 𝑣 − 1
𝑣 + 3
46. 𝑓 + 3
𝑓 − 8
47. 𝑠(𝑟𝑠−1)(𝑟𝑠−3)
3−𝑠
48. 4𝑑 + 1
3𝑑 + 2
49. 𝑥 + 1
𝑥
50. 𝑎2 + 𝑎𝑏 + 𝑏2
6𝑐
51. 4
𝑚 + 𝑛
52. 2𝑚
53. 2(3𝑥 − 1)
𝑥2(𝑥 + 1)
54. 𝑙2(3𝑘 − 𝑙)
2𝑘 + 𝑙
55. 𝑥 − 4
𝑥2𝑦2(𝑥 − 2)
56. −2𝑘2
2𝑠 + 3
57. − 𝑝 − 2
2(𝑝 + 3)
58. −1
4𝑥 + 1
Alg II - Rationals ~16~ NJCTL.org
Multiplying & Dividing Rational Expressions Class Work:
59. 22𝑥2
3
60. 3𝑏𝑐
2𝑏5𝑐
61. 𝑓−2
𝑑+𝑒
62. (𝑔+2)(𝑔−4)
2(𝑔+1)(𝑔+3)
63. 𝑔 + 2
ℎ − 1
64. 7
45
65. 9𝑗2
2
66. 𝑘−1
2𝑘+3𝑙
67. 𝑚−2
(𝑚+3)(𝑚+2)
68. 1
2(𝑛+2)
69. (𝑝+1)2
(𝑝+5)(𝑝+2)2
70. 9𝑟3
4𝑞13
71. 𝑥2 + 3𝑥 + 9
2
72. −3𝑚 − 1
𝑚 + 3
73. 𝑛(𝑚 + 𝑛)
74. 1
3
75. 2𝑚 − 3
2(𝑚 + 3)
76. −𝑥 − 2
(𝑥 + 3)(3𝑥 −1)
77. 3(2𝑥+1)
𝑥+1
Multiplying & Dividing Rational Expressions Homework
78. 33
4𝑜𝑟 8
1
4
79. 9𝑐4
𝑏3
80. 𝑓+2
(𝑑+𝑒)2
81. 𝑔+5
2(𝑔+1)
82. (ℎ+3)(ℎ+7)
(𝑔−4)(ℎ−3)
83. 1
12
84. 9𝑗
2
85. 𝑘+1
𝑘−1
86. 𝑚 − 3
(𝑚 + 4)(𝑚 −1)
87. (𝑛+1)(𝑛+2)
𝑛−2
88. 𝑝+2
(𝑝+3)2
89. 5
3𝑟10𝑞7
90. 2𝑥 + 𝑦
2
91. 2𝑚 − 𝑛
𝑛(3𝑚 − 𝑛)
92. 10
𝑐(𝑎 + 𝑏)
93. (2𝑥 − 5)(3𝑥 − 5)
(4𝑥 − 1)(2𝑥 + 3)
94. −1
𝑥 + 5
95. −𝑚(𝑚 + 2)2
96. 1
2
Adding & Subtracting Rational Expressions Classwork
97. 4
𝑥
98. 𝑦
99. 𝑧−1
2
100. 1 101. 3 102. -1
103. 5𝑥−4
(𝑥+4)(𝑥−4)
104. −2𝑥2−6𝑥+5
(𝑥−3)(𝑥+3)
105. 11𝑥+2
(𝑥+2)2(𝑥−2)
106. 6𝑥−17
(𝑥−4)(𝑥−3)
107. 6𝑥−19
(𝑥−2)(𝑥−3)(𝑥+5)
108. 3𝑥2+12𝑥𝑦 − 𝑥 + 4
𝑥2 + 4𝑦
109. −2𝑥2 + 7
𝑥(3𝑥−1)
110. −𝑚 + 14
𝑚 − 6
111. −5𝑥(𝑥 − 1)
𝑥 − 4
112. 2𝑥3− 2𝑥2+ 14𝑥 − 9
(2𝑥 − 3)(𝑥 − 1)(𝑥 + 2)
113. −15𝑦2+ 6𝑦 + 9
3𝑦 + 2
Alg II - Rationals ~17~ NJCTL.org
Adding and Subtracting Rational Expressions Homework
114. 4𝑥
115. 𝑦
116. 5𝑧−2
3
117. 1
118. 2(𝑣+4)
2𝑣+3
119. 6(𝑢−2)
2𝑢−1
120. 2(3𝑡+1)
(𝑡+1)(3𝑡+1)
121. −2𝑥2+4𝑥+5
(𝑥−2)(𝑥−3)
122. 11𝑥+3
(𝑥+3)2(𝑥−3)
123. 3(2𝑥+1)
(𝑥+3)(𝑥−6)
124. 6𝑥+25
(𝑥−1)(𝑥+4)(𝑥−3)
125. 10𝑥 − 5
3𝑥 − 2
126. 14𝑥 − 5
2𝑥(𝑥 − 3)
127. −7𝑝 + 3
𝑝 − 4
128. −2(7𝑏 − 3)
2𝑏 − 3
129. −18𝑚2+ 33𝑚 + 3
(4𝑚 − 1)(4𝑚 + 1)(2𝑚 − 1)
130. −3𝑦3 + 6𝑦2 −10𝑦 + 7
(2𝑦 + 1)(3𝑦 − 1)(𝑦 − 1)
Solving Rational Equations Class Work
131. −13
132. 13
4𝑜𝑟 3
1
4
133. −2
134. 4
5,
3
2
135. −17
5
136. −3
7
137. −16
138. 5±√69
2
139. 19±√2811
4
140. No Solution
141. −11
18
142. 2
143. 11 ± √73
4
Solving Rational Equations Homework
144. 13
3
145. 2 146. 7
147. 3(5±√105)
20
148. 12
5
149. 38
11
150. 27
5
151. No Solution
152. −7±𝑖√55
4
153. ±2
154. 2
7
155. −3 ± √15
2
156. 3
8
Alg II - Rationals ~18~ NJCTL.org
Graphing Rational Equations Classwork:
157. 𝑓(𝑥) =2
𝑥−1
x-intercepts: None y-intercepts: 𝑦 = 2
Holes: None
Vertical asymptotes: 𝑥 = 1
Horizontal asymptotes: 𝑦 = 0
158. 𝑔(𝑥) =−3
𝑥+2
x-intercepts: None
y-intercepts: 𝑦 = −1.5
Holes: None
Vertical asymptotes: 𝑥 = −2
Horizontal asymptotes: 𝑦 = 0
159. ℎ(𝑥) =𝑥+1
𝑥2−1
x-intercepts: None
y-intercepts: 𝑦 = −1
Holes: 𝑥 = −1
Vertical asymptotes: 𝑥 = 1 Horizontal asymptotes: 𝑦 = 0
160. 𝑓(𝑥) =𝑥−1
(𝑥−1)(𝑥+2)
x-intercepts: None
y-intercepts: 𝑦 =1
2
Holes: 𝑥 = 1
Vertical asymptotes: 𝑥 = −2
Horizontal asymptotes: 𝑦 = 0
Alg II - Rationals ~19~ NJCTL.org
161. 𝑔(𝑥) =𝑥2+5𝑥+6
𝑥2+3𝑥+2
x-intercepts: 𝑥 = −3
y-intercepts: 𝑦 = 3
Holes: 𝑥 = −2
Vertical asymptotes: 𝑥 = −1
Horizontal asymptotes: 𝑦 = 1
162. ℎ(𝑥) =𝑥2−𝑥−6
𝑥2−5𝑥+6
x-intercepts: 𝑥 = −2
y-intercepts: 𝑦 = −1
Holes: 𝑥 = 3
Vertical asymptotes: 𝑥 = 2
Horizontal asymptotes: 𝑦 = 1
Graphing Rational Equations Homework:
163. 𝑓(𝑥) =2
𝑥+3
x-intercepts: None
y-intercepts: 𝑦 =2
3
Holes: None
Vertical asymptotes: 𝑥 = 3
Horizontal asymptotes: 𝑦 = 0
164. 𝑔(𝑥) =−3
𝑥−4
x-intercepts: None
y-intercepts: 𝑦 =3
4
Holes: None
Vertical asymptotes: 𝑥 = 4
Horizontal asymptotes: 𝑦 = 0
Alg II - Rationals ~20~ NJCTL.org
165. 𝑓(𝑥) =𝑥+2
(𝑥−1)(𝑥+2)
x-intercepts: None
y-intercepts: 𝑦 = −1
Holes: 𝑥 = −2
Vertical asymptotes: 𝑥 = 1
Horizontal asymptotes: 𝑦 = 0
166. ℎ(𝑥) =𝑥−2
𝑥2−4
x-intercepts: None
y-intercepts:𝑦 =1
2
Holes: 𝑥 = 2
Vertical asymptotes: 𝑥 = −2
Horizontal asymptotes: 𝑦 = 0
167. 𝑔(𝑥) =𝑥2+9𝑥+18
𝑥2+7𝑥+6
x-intercepts: 𝑥 = −3
y-intercepts: 𝑦 = 3
Holes: 𝑥 = −6
Vertical asymptotes: 𝑥 = −1
Horizontal asymptotes: 𝑦 = 1
168. ℎ(𝑥) =𝑥2+5𝑥−14
𝑥2+6𝑥−7
x-intercepts: 𝑥 = 2
y-intercepts: 𝑦 = 2
Holes: 𝑥 = −7
Vertical asymptotes: 𝑥 = 1
Horizontal asymptotes: 𝑦 = 1
Alg II - Rationals ~21~ NJCTL.org
Unit Review - Multiple Choice
1. B.
2. C.
3. A.
4. A.
5. C.
6. A.
7. B.
8. D.
9. D.
10. C.
11. D.
12. A.
Extended Response
1. a. (x - 4)(x +3)
x + 4
b. x3 + 4x2 -16x-64
c. x+3
2.
x-intercepts: (2, 0)
y-intercepts: (0,1
2)
Hole: 𝑥 = −3
Vertical Asymptote: 𝑥 = 2
Horizontal Asymptote: 𝑦 = −1
4