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MULTIPLICATION AND DIVISION OF RATIONAL EXPRESSIONS

Definitions:

Rational Expression : is the quotient of two polynomials. For example,

are all rational expressions.

Lowest terms : A rational expression is in lowest terms when the numerator and denominator
contain no common factors .

Reciprocal : The reciprocal of a rational expression a/b is given by b/a . To find the reciprocal, we invert
(or flip) the rational expression.

Important Properties:
• Fundamental Property of Rational Numbers: If a/b is a rational number and c is any nonzero
real number , then

We use this property to write rational expressions in lowest terms.

• To write a rational expression in lowest terms : Factor both the numerator and denominator
completely. Apply the Fundamental Property of Rational Numbers to eliminate the common factors.

• To multiply rational expressions: Factor all numerators and denominators as much as possible.
Apply the Fundamental Property of Rational Numbers to eliminate the common factors. Multiply
remaining factors.

• To divide rational expressions: Invert the second rational expression and multiply. In other
words, multiply the first rational expression by the reciprocal of the second .

• If the numerator and denominator of a rational expression are opposites then the answer is -1. This
is because if we factor out a -1 from either the numerator or denominator, we have

Common Mistakes to Avoid:

• Remember that only common factors can be divided out. For example,

however, and

• To multiply (or divide) rational expressions, you do NOT need a common denominator.

• x+y and y +x are NOT opposites of one another. Recall that is does not matter the order in which
we add terms together. As a result,

PROBLEMS

Simplify each expression in lowest terms.

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