1. Rational Expressions Much of the terminology and many of the techniques for the arithmetic of fractions of real numbers carry over to algebraic fractions, which are the quotients of algebraic expressions. In particular, the quotient of two polynomials is referred to as a rational expression. The rules for multiplying and dividing rational expressions are the same as those for multiplying and dividing fractions of real numbers. Do you recall what they are? To simplify a rational expression, use the cancellation principle:

2. Factoring and Cancellation Simplification of a rational expression is often a two-step process: (1) Factor, and (2) Cancel. Problem. Simplify Solution. (1) Factor numerator and denominator: (2) Cancel common factors: Warning!!! Only multiplicative factors can be cancelled.

3. Adding and Subtracting Rational Expressions When two rational expressions have the same denominator, the addition and subtraction rules are: To add or subtract rational expressions with different denominators, we must first rewrite each rational expression as an equivalent one with the same denominator as the others. Example. Add the rational expressions: Since the denominators are different, we convert each expression to an equivalent expression with denominator 6.

4. Least Common Denominator Although any common denominator will do for adding rational expressions, we will concentrate on finding the least common denominator, or LCD, of two or more rational expressions. The LCD is found by a 3-step process: (1) Factor the denominator of each fraction, (2) Find the highest power (final factor) to which each factor occurs, and (3) The LCD is the product of the final factors. Example. Find the LCD:

5. Addition of Rational Expressions Addition of rational expressions is a 3-step process: (1) Find the LCD. (2) Write each expression as an equivalent expression which has denominator equal to the LCD. (3) Add the rational expressions from Step 2. Example.

6. Complex Fractions We want to simplify a complex fraction, which is a fractional form with fractions in the numerator or denominator or both. Simplifying a complex fraction is a 2-step process. (1) Find the LCD of all fractions in the numerator and denominator. (2) Multiply both numerator and denominator by the LCD. Example. (1) The LCD is ab. (2) The result is

7. Summary of Rational Expressions; We discussed: Rational expressions The cancellation principle Addition of rational expressions with the same denominator Least common denominator (LCD) and how to find it Addition of general rational expressions as a 3-step process Simplification of complex fractions as a 2-step process