Section 8.3: Adding and Subtracting Rational

Expressions

8.3 Lecture Guide: Adding and Subtracting Rational Expressions

Objective: Add and Subtract rational expressions.

Adding or Subtracting Rational Expressions with the Same Denominator

Algebraically

Verbally

To add two rational expressions with the same denominator, add the numerators and use this common denominator.

If A, B and C are real polynomials and 0C then A B A B

C C C

9 5

28 28 28

28

x x x

x

Algebraic Example

Adding or Subtracting Rational Expressions with the Same Denominator

Algebraically

Verbally To subtract two rational expressions with the same denominator, subtract the numerators and use this common denominator.

If A, B and C are real polynomials and 0C then

Algebraic Example

A B A B

C C C

9 5

28 28 28

28

x x x

x

1.2 3 2 3

2 6

4 4

x x

x y x y

Perform the indicated operations, and reduce the results to lowest terms. Assume the variables are restricted to values that prevent division by zero.

2. 3 2 3 2

6 2 8

5 5

x y x y

x y x y

3.3 8 2 5

4 12 4 12

x x

x x

4.2

2 2

3 6 10

4 4

x x x

x x

5.3 5

2 1 1 2

x

x x

6.2 6

3 3

x

x x

Finding the LCD of Two or More Rational Expressions

Algebraic Example Verbally 1. Factor each denominator completely, including constant factors. Express repeated factors in exponential form.

2. List each factor to the highest power to which it occurs in any single factorization.

3. Form the LCD by multiplying the factors listed in Step 2.

Determine the LCD of

3 2

8

10x yand

2 4

5

12x y3 2

2 4

10

12

x y

x y

LCD

LCD

Determine the LCD of each pair of expressions.

7.2

3 1

5 20

x

x

and 3

7 14

x

x

Determine the LCD of each pair of expressions.

8. and 3 2

2 5

4 4

x

x x

2

5 1

3 6 3

x

x x

Adding (Subtracting) Rational Expressions:

Algebraic Example Verbally

1. Express the denominator of each rational expression in factored form, and then find the LCD.

2. Convert each term to an equivalent rational expression whose denominator is the LCD.

3 2 2 4

8 5

10 12x y x y

Adding (Subtracting) Rational Expressions:

Algebraic Example Verbally

3. Retaining the LCD as the denominator, add (subtract) the numerators to form the sum (difference).

4. Reduce the expression to lowest terms.

9.1 2

5 4x x

Perform the indicated operations, and reduce the results to lowest terms. Assume the variables are restricted to values that prevent division by zero.

10.3 1

2 6x x

11. 2 2

2 1

2 35 3 24 45x x x x

12. 2 2

3 2

4 5 2 2x x x

13.2

2 3 18

3 3 9

x x

x x x

14. 2

77 3 7

30 5 6x x x x