1. Section R.6
Rational Expressions

2. R.6 Rational Expressions
Determine the domain of a rational expression.
Simplify rational expressions.
Multiply, divide, add, and subtract rational expressions.
Simplify complex rational expressions.

3. Domain of Rational Expressions
The domain of an algebraic expression is the set of all real numbers for which the expression is defined. Division by zero is undefined.
Example: Find the domain of
Solution: We solve the equation x – 3 = 0 to determine the numbers that are not in the domain:
Since the denominator is 0 when x = 3, the domain of 2/(x – 3) is the set of all real numbers except 3.

4. Simplifying, Multiplying, and Dividing Rational Expressions
To simplify rational expressions, we use the fact that

5. Simplifying, Multiplying, and Dividing Rational Expressions
Solution

6. Adding and Subtracting Rational Expressions
When rational expressions have the same denominator, we can add or subtract the numerators and retain the common denominator. If the denominators are different, we must find equivalent rational expressions that have a common denominator.
To find the least common denominator of rational expressions, factor each denominator and form the product that uses each factor the greatest number of times it occurs in any factorization.

7. Example Subtract: Solution:
The LCD is (2x – 1)(x – 1)(2), or 2(2x – 1)(x – 1).

8. Example continued

9. Complex Rational Expressions
A complex rational expression has rational expressions in its numerator or its denominator or both.
To simplify a complex rational expression:
Method 1. Find the LCD of all the denominators within the complex rational expression. Then multiply by 1 using the LCD as the numerator and the denominator of the expression for 1.
Method 2. First add or subtract, if necessary, to get a single rational expression in the numerator and in the denominator. Then divide by multiplying by the reciprocal of the denominator.

10. Example
Simplify:
Using Method 1, the LCD of the four rational expressions in the numerator and the denominator is a3b3.

11. Example continued