AA 8.4 & 8.5 Combined Notes
Name _____________________________________
Simplifying rational expressions usually requires two steps:
𝑎𝑐
𝑎𝑏
4𝑥 + 12
2
𝑥 − 2𝑥 − 15
1.
Factor the numerator and denominator.
2.
Cancel out anything that is both in the numerator and denominator.
𝑥 2 − 2𝑥 − 15
𝑥2 − 9
Multiplying Rational expressions:
1.
𝑎 𝑐
×
𝑏 𝑑
Factor numerators and denominators
2.
Multiply numerators and denominators
3.
Change signs as needed by factoring out “-1”
4.
Divide out common factors
5.
Simplify completely
5𝑥 2
6𝑥𝑦 3
×
6𝑥𝑦 2 10𝑥𝑦 3
3𝑥 − 3𝑥 2
𝑥 2 + 𝑥 − 20
×
𝑥 2 + 4𝑥 − 5
3𝑥
5𝑥 2 10𝑥
÷ 3
𝑦2
𝑦
𝑥2 + 𝑥 − 6 𝑥 + 3
÷
𝑥2 + 𝑥
𝑥
Divide Rational expressions:
𝑎 𝑐
÷
𝑏 𝑑
1.
Factor numerators and denominators and cancel
2.
Multiply by the reciprocal of the second fraction
Adding and subtracting rational expressions with common “like” denominator
𝑎 𝑏
+
𝑐 𝑐
7 3𝑥 2
+
4𝑥 4𝑥
𝑥 2 − 3𝑥
2
+ 2
2
𝑥 +1 𝑥 +1
𝑎 𝑏
−
𝑐 𝑐
2𝑥
5
−
𝑥+6 𝑥+6
𝑥2 + 𝑥 𝑥 + 4
−
𝑥+2 𝑥+2
1.
Add the numerator and “keep” the denominator
2.
Factor out and simply the answer
Adding and subtracting rational expressions with Uncommon “Unlike” denominator
𝑎 𝑏
+
𝑐 𝑑
7 3𝑥 2
+
4𝑥
3
7
𝑥
+ 2
2
9𝑥
3𝑥 + 3𝑥
𝑎 𝑏
−
𝑐 𝑑
2𝑥
5
−
𝑥 + 5 6𝑥
𝑥+2
−2𝑥 − 1
− 2
2𝑥 − 2 𝑥 − 4𝑥 + 3
1.
Find the least common multiple LCM of each denominator LCD.
Hint: the simplest way is to multiply the denominators together but this may be the “messiest” way.
FACTOR the denominators if you can this is the ___________________________
2.
Multiply by “some form of 1” to every term to make the denominators the same
Examples:
Simplify the Rational Expression
18x
4x
1. 12
2. 36
18 x 2 16 x 3
2x
3.
3( x  1)
4. 6( x  1)
2(3x  2)
5. (3 x  2)
3( x  4)( x  2)
9( x  2)
6.
(2 x  6)
4
7.
3x 2
(2 x 3  x 2 )
8.
(2 x 2  x)
(3x 2  2 x)
9.
x 2  3x  2
2
10. x  5 x  6
x 2  2x  3
2
11. x  7 x  12
x 2  16
12. 3 x  12
( x  3)
2
13. x  5 x  6
x 3  27
3
2
14. x  3x  9 x
3x 2  3x  6
x2  4
15.
Simplify
7x the Rational Expression
a. 21
 5x
b. 10
2
c. 2 x  4
36 x 4
7
d. 42 x
5x
2
e. x  3 x
2x 2  x
4x
f.
x2 1
g. 6 x  6
4 x  12
2
h. x  9
x 2  3x  10
2
i. x  5 x  6
x2
3
j. x  8
x 3  x 2  12 x
x3  9x
k.
x 2  6x  8
(4  x)
l.
x 3  3x 2  2 x  6
x 3  27
m.
2 x 2  4 x  30
2
n. 2 x  8 x  6
3x 2  9 x  12
2
o. 2 x  26 x  72
Examples: Multiply or Divide the Rational Expression
3x 10
8x 6


x
a. 6
b. 5 12 x
5 x 2 10 x 3

21
d. 7
1
3

2
c. 5 x 20 x
x4
6

2( x  4)
e. 3
x3 x3
 2
f. x  3 x  9
x3
4( x  3)

g. 6(2 x  1) 3(2 x  1)
3
x 2  3x
 2
h. 4 x  8 x  x  6
x 2  25 36

12
x5
i.
x3
 (16 x 2  4 x  1)
3
j. 64 x  1
x
9 x 2  25
 (3x  5) 
x5
k. x  5
x 2  11x
x2  4
 (3x 2  6 x) 
x2
x  11
l.
Exercises:
8 x 10 Multiply or Divide the7Rational
x 10 Expression
 2

x
1. 10 x
2. 5
x3 x3

5
4.  4
4 x 2 8x 3
 5
3x
3. 9
x 2  25
3

12
x5
5.
x4
 (3x  3)
6. x  5 x  4
3x
9x

7. 2 x  10 3x  15
x 2  16 48

12
x4
8.
x3
 (5 x  10)
9. x  5 x  6
x3
x3

10. x  3x  10 8 x  16
3x 2  x  2
2x

2
11. x  3x  2 x  2
x 2  7 x  10
 (8 x  16)
2
12. x  2 x  35
x 2  5 x  4 x 2  3x  4

x 2  3x
2x 2  6x
13.
2
2
2
x 2  11x  30
 ( x  6)
x 2  7 x  12
x
x2 1
 ( x  1) 
x3
14. x  3
15.
x 2  11x
x2  4
 (3x 2  6 x) 
x2
x  11
16.
x 2  x  12 x 3  3x 2 4 x  1
 3

8x 2
8x  2 x 2 x  2
17.
( x  5) 
Find the
14 least7 common denominator
4
,
1. 4( x  1) 4 x
2.
21x
2
,
x
3x  15 x
2
Perform
the indicated operation(s)
7 11
4x
3and simplify
5.
6x

6x
6.
x 1

x 1
5x  2 3 9 x
, ,
2
3. 4 x  1 x 2 x  1
3x  1
3
, 2
4. x( x  7) x  6 x  7
x
5
 2
2
7. x  5 x x  5 x
5x 2
5x

8. x  8 x  8
6
2

2
5x
9. 4 x
7
x3

6x
10. 6( x  2)
10
2

2
11. x  5 x  14 x  7
4x 2
10

12. 3 x  5 x  8
x2  x  3
3x

2
13. x  12 x  32 x  8
4x
5
4


14. x  1 2 x  3 x
20
x 1
1
7

15. 4 x  1
1
5

4 x  3 3(4 x  3)
x
4x  3
16.
4
2

x 9 x3
1
1

17. x  3 x  3
2