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Sums and Differences of
Rational Expressions
Objective: to add and subtract rational expressions
Rules for adding and subtracting fractions
1. Find the least common denominator (LCD) of the fractions (that is the least common multiple of the denominators.)
2. Express each fraction as an equivalent fraction with the LCD as the denominator
3. Add or subtract and then simplify.
Examples
Simplify
1. The LCM of 5 = 5 • 1 and 15 = 5 • 3 is 15
2. multiplying by 1 will not change a number
3.
1 7
5 15
3 1 7
3 5 15
3 7 10 5 2 2
15 15 15 5 3 3
Example with monomials
Simplify
1. The LCM of 3x2y, 12x, and 8y2 is 24x2y2
2.
3.
2 2
2 1 3
3 12 8x y x y
2
2 2 2
8 2 2 1 3 3
8 3 2 12 3 8
y xy x
y x y xy x x y
2 2
2 2 2 2 2 2 2 2
16 2 9 16 2 9
24 24 24 24
y xy x y xy x
x y x y x y x y
Example with Polynomials
Simplify
1. Factor
2. The LCM is (x + 3)(x – 3)(x – 1)
3.
2 2
1
9 4 3
x
x x x
1
3 3 3 1
x
x x x x
1 3 1
1 3 3 3 3 1
x x x
x x x x x x
1 3 1
1 3 3 3 3 1
x x x
x x x x x x
2 3
3 3 1 3 3 1
x x x
x x x x x x
2 3
3 3 1 3 3 1
x x x
x x x x x x
2 2 3
3 3 1
x x
x x x
Be careful, its not done yet
factor
2 2 3
3 3 1
x x
x x x
3 1
3 3 1
x x
x x x
1
3 1
x
x x
Try These!Simplify
1.
2.
3.
4.
1.
2.
3.
4.
5 1 7
6 3 18
2
1 1x x
x x
22
1
1 1
x
x x
2
2
9 3
3 9 3
x x
x x x
7
9
2
1 1x x
x
6
3x End Show
2
2
2 1
( 1) ( 1)
x x
x x
5 1 7
6 3 18
3 5 6 1 7
3 6 6 3 18
15 6 7
18 18 18
14 7
18 9
Try These!
2
1 1x x
x x
2
2 2 2
1 1 1x x x x x x
x x x x x
2
2 2
1x x x
x x
2
2
1x
x
Try These!
22
1
1 1
x
x x
2
1
( 1)( 1) 1
x
x x x
22
( 1) 1( 1)
( 1) ( 1) 1 ( 1)
x x x
x x x x
2
22
1
( 1) ( 1) 1 ( 1)
x x x
x x x x
2 2
2 2
2 1 2 1
( 1) ( 1) ( 1) ( 1)
x x x x
x x x x
Try These!
2
2
9 3
3 9 3
x x
x x x
23 39 3
3 3 3 3 3 3
x xx x
x x x x x x
2 23 9 3 9
3 3 3 3 3 3
x x x x
x x x x x x
2 23 9 3 9
3 3 3 3 3 3
x x x x
x x x x x x
2 23 9 3 9
3 3 3 3 3 3
x x x x
x x x x x x
6 36 18
3 3 3 3
6
3
xx
x x x x x
Try These!
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