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# Rational Expressions

Recall:

A rational number is a number which can be written as a fraction. The root word is ratio.

New Stuff:

• It stands to reason, therefore, that anything called rational in mathematics is likely to refer to fractions or, at least, a fractional form.

Definition (Rational Expression)

A rational expression is a “fraction” in which both the top and the bottom are polynomials.

• Unlike fractions, we must be concerned about variables. Specifically, if a value of a variable would make the bottom of the rational expression equal 0, then that value must be discarded (from the list of possible values of the variable).

Procedure: (Determining the Numbers for which a Rational Expression is Undefined)

Set the denominator equal to zero and solve. Any solutions to this equation are numbers that make the rational expression undefined (since you would then be dividing by zero).

• Simplifying a rational expression is the same as “reducing to lowest terms”. In other words, you cancel any common factors between the top and the bottom. To do this, you will need to factor the top and bottom COMPLETELY.

Procedure: (Simplifying a Rational Expression)

1. Factor the top and bottom of the rational expression completely.

2. “Cancel” any common factors between the top and the bottom.

IMPORTANT: You MUST determine for what numbers the expression is undefined BEFORE you reduce the expression to lowest terms.

Examples:

Simplify. For expressions with only one variable, determine for what numbers the expression is undefined.

DO NOT CANCEL PIECES OF THE SUMS! YOU MUST FACTORE BEFORE YOU BEGIN CANCELLING!

Watch out for factors that are opposites of one another. If you follow my earlier advice and always factor out a negative number if the leading coefficient is negative, then you should have no real problems with this.

Examples: