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8-2 Study Guide and Intervention
Adding and Subtracting Rational Expressions
LCM of Polynomials To find the least common multiple of two or more polynomials, factor each expression.
The LCM contains each factor the greatest number of times it appears as a factor.
Example 1: Find the LCM of 16𝒑𝟐 𝒒𝟑r, 40p𝒒𝟒 𝒓𝟐,
Example 2: Find the LCM of 3𝒎𝟐 – 3m – 6 and
16𝑝2 𝑞3 r = 24 · 𝑝2 · 𝑞3 · r
3𝑚 2 – 3m – 6 = 3(m + 1)(m – 2)
40p𝑞4 𝑟 2 = 23 · 5 · p · 𝑞4 · 𝑟 2
4𝑚 2 + 12m – 40 = 4(m – 2)(m + 5)
and 15𝒑𝟑 𝒓𝟒.
4𝒎𝟐 + 12m – 40.
15𝑝3 𝑟 4 = 3 · 5 · 𝑝3 · 𝑟 4
LCM
4
3
LCM = 12(m + 1)(m – 2)(m + 5)
4
=2 ·3·5·𝑝 ·𝑞 ·𝑟
4
= 240𝑝3 𝑞4 𝑟 4
Exercises
Find the LCM of each set of polynomials.
1. 14a𝑏2 , 42b𝑐 3 , 18𝑎2 c
2. 8cd𝑓 3 , 28𝑐 2 f, 35𝑑4 𝑓 2
3. 65𝑥 4 y, 10𝑥 2 𝑦 2 , 26𝑦 4
4. 11m𝑛5 , 18𝑚 2 𝑛3 , 20m𝑛4
5. 15𝑎4 b, 50𝑎2 𝑏2 , 40𝑏8
6. 24𝑝7 q, 30𝑝 2 𝑞2 , 45p𝑞3
7. 39𝑏2 𝑐 2 , 52𝑏4 c, 12𝑐 3
8. 12x𝑦 4 , 42𝑥 2 y, 30𝑥 2 y3
9. 56st𝑣 2 , 24𝑠 2 𝑣 2 , 70𝑡 3 𝑣 3
10. 𝑥 2 + 3x, 10𝑥 2 + 25x – 15
11. 9𝑥 2 – 12x + 4, 3𝑥 2 + 10x – 8
12. 22𝑥 2 + 66x – 220, 4𝑥 2 – 16
Chapter 8
11
Glencoe Algebra 2
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13. 8𝑥 2 – 36x – 20, 2𝑥 2 + 2x – 60
14. 5𝑥 2 – 125, 5𝑥 2 + 24x – 5
15. 3𝑥 2 – 18x + 27, 2𝑥 3 – 4𝑥 2 – 6x
16. 45𝑥 2 – 6x – 3, 45𝑥 2 – 5
17. 𝑥 3 + 4𝑥 2 – x – 4, 𝑥 2 + 2x – 3
18. 54𝑥 3 – 24x, 12𝑥 2 – 26x + 12
8-2 Study Guide and Intervention (continued)
Adding and Subtracting Rational Expressions
Add and Subtract Rational Expressions To add or subtract rational expressions, follow these steps.
Step 1 Find the least common denominator (LCD). Rewrite each expression with the LCD.
Step 2 Add or subtract the numerators.
Step 3 Combine any like terms in the numerator.
Step 4 Factor if possible.
Step 5 Simplify if possible.
𝟔
𝟐
Example: Simplify 𝟐𝒙𝟐 + 𝟐𝒙− 𝟏𝟐 – 𝒙𝟐 − 𝟒.
6
2
2𝑥 2 + 2𝑥 – 12
– 𝑥2 − 4
6
2
= 2(𝑥 + 3)(𝑥 − 2) – (𝑥 − 2)(𝑥 + 2)
6(𝑥 + 2)
2 · 2(𝑥 + 3)
= 2(𝑥 + 3)(𝑥 − 2)(𝑥 + 2) – 2(𝑥 + 3)(𝑥 − 2)(𝑥 + 2)
6(𝑥 + 2) − 4(𝑥 + 3)
= 2(𝑥 + 3)(𝑥 − 2)(𝑥 + 2)
6𝑥 + 12 − 4𝑥 − 12
= 2(𝑥 + 3)(𝑥 − 2)(𝑥 + 2)
2𝑥
= 2(𝑥 + 3)(𝑥 − 2)(𝑥 + 2)
𝑥
= (𝑥 + 3)(𝑥 − 2)(𝑥 + 2)
Factor the denominators.
The LCD is 2(x + 3)(x – 2)(x + 2).
Subtract the numerators.
Distribute.
Combine like terms.
Simplify.
Exercises
Simplify each expression.
1.
−7𝑥𝑦
3𝑥
Chapter 8
+
4𝑦 2
2𝑦
2.
2
𝑥–3
–
1
𝑥–1
12
Glencoe Algebra 2
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4𝑎
15𝑏
3
3. 3𝑏𝑐 – 5𝑎𝑐
3𝑥 + 3
4𝑥 + 5
4. 𝑥 + 2 + 3𝑥 + 6
𝑥 −1
5. 𝑥 2 + 2𝑥 + 1 + 𝑥 2 – 1
4
5𝑥
6. 4𝑥 2 − 4𝑥 + 1 – 20𝑥 2 − 5
8-2 Skills Practice
Adding and Subtracting Rational Expressions
Find the LCM of each set of polynomials.
1. 12c, 6𝑐 2 d
3. 2x – 6, x – 3
5. 𝑡 2 – 25, t + 5
2. 18𝑎3 b𝑐 2 , 24𝑏2 𝑐 2
4. 5a, a – 1
6. 𝑥 2 – 3x – 4, x + 1
Simplify each expression.
3
5
3
7. 𝑥 + 𝑦
9.
2𝑐 − 7
3
Chapter 8
5
8. 8𝑝2 𝑟 + 4𝑝2 𝑟
+4
2
5
10. 𝑚2𝑝 + 𝑝
12
Glencoe Algebra 2
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
11.
12
5𝑦 2
–
2
2
12.
5𝑦𝑧
3
2
𝑘
𝑥
2𝑛 + 2
21. 𝑛 – 3 + 𝑛2 − 2𝑛 – 3
Chapter 8
3𝑡
5
4𝑧
𝑧+4
18. 𝑧 – 4 + 𝑧 + 1
19. 𝑥 2 + 2𝑥 + 1 + 𝑥 + 1
𝑛
2
16. 2 – 𝑥 + 𝑥 – 2
17. 𝑘 – 𝑛 – 𝑛 – 𝑘
1
3
4ℎ 2
14. 3𝑏 + 𝑑 – 3𝑏𝑑
15. 𝑤 – 3 – 𝑤 2 – 9
𝑘
+
5
13. 𝑎 + 2 – 2𝑎
3
7
4𝑔ℎ
20.
2𝑥 + 1
𝑥–5
3
4
– 𝑥 2 − 3𝑥 – 10
2
22. 𝑦2 + 𝑦 – 12 – 𝑦2 + 6𝑦 + 8
12
Glencoe Algebra 2