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Addition, Subtraction, and Least Common Denominators

• Addition When Denominators Are the Same

• Subtraction When Denominators Are the Same

• Least Common Multiples and Denominators

6.3

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The Sum of Two Rational Expressions

To add when the denominators are the same, add the numerators and keep the common denominator:

.A C A C

B B B

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Example

Add. Simplify the result, if possible.

a) b)

c) d)

Solution

5 6 w

w w

4 3 2

7 7

x x

x x

2 2

9 3

36 36

x

x x

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The Difference of Two Rational Expressions

To subtract when the denominators are the same, subtract the second numerator from the first and keep the common denominator:

.A C A C

B B B

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Example

Subtract and, if possible, simplify:

a) b)

Solution

5 4

3 3

x x

x x

2 30

6 6

x x

x x

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Least Common Multiples and Denominators

To add or subtract rational expressions that have different denominators, we must first find equivalent rational expressions that do have a common denominator.

The least common multiple must include the factors of each number, so it must include each prime factor the greatest number of times that it appears in any factorizations.

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To Find the Least Common Denominator (LCD)1. Write the prime factorization of each denominator.2. Select one of the factorizations and inspect it to see

if it completely contains the other factorization.a) If it does, it represents the LCM of the denominators.b) If it does not, multiply that factorization by any

factors of the other denominator that it lacks. The final product is the LCM of the

denominators.

The LCD is the LCM of the denominators. It should contain each factor the greatest number of times that it occurs in any of the individual factorizations.

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Example

Find the LCD of Solution

2 3

7 3 and .

6 4x x

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Example

For each pair of polynomials, find the least common multiple.a) 16a and 24bb) 24x4y4 and 6x6y2

c) x2 4 and x2 2x 8Solution

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Example

For each group of polynomials, find the least common multiple.a) 15x, 30y, 25xyz b) x2 + 3, x + 2, 7Solution

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Example

Find equivalent expressions that have the LCD:

Solution

2 2

3 5;

4 2 8

x x

x x x