Section 9.3 Adding and Subtracting Rational

Expressions

9.3 Lecture Guide: Adding and Subtracting Rational Expressions

Objective 1: Add and Subtract rational expressions.

Adding Rational Expressions with the Same Denominator

Algebraically Verbally Algebraic Examples

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

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

0,C

.A B A BC C C

9 528 28 28

28

x x x

x

Subtracting Rational Expressions with the Same Denominator

Algebraically Verbally Algebraic Examples

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

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

0,C

A B A BC C C

9 528 28 28

28

x x x

x

Perform the indicated operations, and reduce the results tolowest terms. Assume the variables are restricted to valuesthat prevent division by zero.

1. 2 3 2 3

2 64 4x xx y x y

Perform the indicated operations, and reduce the results tolowest terms. Assume the variables are restricted to valuesthat prevent division by zero.

2. 3 2 3 2

6 2 85 5x y x yx y x y

Perform the indicated operations, and reduce the results tolowest terms. Assume the variables are restricted to valuesthat prevent division by zero.

3. 3 8 2 54 12 4 12x xx x

Perform the indicated operations, and reduce the results tolowest terms. Assume the variables are restricted to valuesthat prevent division by zero.

4. 2

2 2

3 6 104 4

x x xx x

Perform the indicated operations, and reduce the results tolowest terms. Assume the variables are restricted to valuesthat prevent division by zero.

5. 3 52 1 1 2xx x

Perform the indicated operations, and reduce the results tolowest terms. Assume the variables are restricted to valuesthat prevent division by zero.

6. 2 63 3x

x x

Finding the LCD of Two or More Rational Expressions

Verbally Algebraic Example

1. _______________ each denominator completely, including constant factors. Express repeated factors in exponential form.

Determine the LCD of

and

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

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

3 2

810x y 2 4

512x y

3 2

2 4

10

12

x y

x y

LCD

LCD

Determine the LCD of each pair of expressions.

7. 2

3 15 20xx

and 37 14x

x

Determine the LCD of each pair of expressions.

8. 3 2

2 54 4xx x

2

5 13 6 3

xx x

and

Adding (Subtracting) Rational Expressions

Verbally Algebraic Example

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 _______________ is the LCD.

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

4. Reduce the expression to lowest terms.

3 2 2 4

7 510 12x y x y

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

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

11.2 2

2 1

2 35 3 24 45x x x x

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

12.2 2

3 1

4 5 1x x x

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

13.2

2 3 18

3 3 9

x x

x x x

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

14.2

77 3 7

30 5 6x x x x

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

15. Total Area of Two Regions Find the sum of the areas of the parallelogram and rectangle.

2

1cm

x

3x cm 4x cm

2

1cm

x

16. One person can load a truck in t hours while a second person would take hours. In 5 hours the fractional

portion of the job done by each is and ,

respectively.

(a) Write a rational function that gives the total amount of work completed by the two workers in 5 hours.

(b) Evaluate this function if .

(c) Use the graph of this function to describe what happens to the total amount of work done as t becomes very large.

3t 5t

53t

12t