# Multiplying and Dividing Rational Expressions

I. Rational Expressions and Functions

A. A rational expression is a polynomial divided by a
nonzero polynomial.

B. A rational function is a function defined by a formula that is a rational
expression.

C. Fractions are rational expressions.

Examples:

What value(s) of x would make the denominators of the
above examples equal to 0?

1)

2)

3)

Why do these values matter ?

C. The domain of a rational function is the set of all real numbers except those

Examples: Find the domain of the following functions. (Easiest to use set-builder notation.)

II. Simplifying Rational Expressions

A. To simplify, reduce , or write in lowest terms means the same thing: the
numerator and denominator

don’t contain any common factors (other than 1 or -1).

B. To simplify:

1) Factor completely the numerator and denominator .

2) Divide both the numerator and denominator by the common factors; that is,
cancel out any

common factors that the numerator and denominator share. NEVER cancel TERMS ;
only FACTORS.

Examples: Simplify.

III. Multiplying Rational Expressions

A. Multiplying rational expressions is the same as multiplying fractions:

B. To multiply rational expressions:

1) Factor completely all numerators and denominators.

2) Divide numerators and denominators by common factors.

3) Multiply the remaining numerators; multiply the remaining denominators.

Examples: Multiply.

IV. Dividing Rational Expressions

A. Dividing rational expressions is the same as dividing fractions:

C. To divide rational expressions:

1) Leave the first rational expression alone; change division to multiplication;
“flip” the divisor.

2) Now it is a multiplication problem. Proceed with multiplication.

Examples: Divide.

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