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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

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.

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.

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

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.

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

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

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