sums and differences of rational expressions objective: to add and subtract rational expressions

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Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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Page 1: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

Sums and Differences of

Rational Expressions

Objective: to add and subtract rational expressions

Page 2: 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.

Page 3: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 4: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 5: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 6: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 7: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 8: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 9: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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!

Page 10: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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!

Page 11: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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!

Page 12: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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

Page 13: Sums and Differences of Rational Expressions Objective: to add and subtract rational expressions

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!