Rational Expressions

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