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TUTORIALS:
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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:
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