Rational expressions: excluded values and simplifying

Fractions with polynomials

Some examples:

Something like \(\frac{x - 3}{4}\) usually isn't called a rational expression: the denominator is just a number, so it's really the polynomial \(\frac{1}{4}x - \frac{3}{4}\).

Rational expressions behave like numerical fractions. You simplify them, find common denominators, multiply and divide. The difference is that instead of prime factorizations of numbers, you use factoring of polynomials.

Excluded values

A fraction with a zero denominator is undefined. So before doing anything else, rule out the values of the variable that make the denominator zero. These are the excluded values; everything else is the domain of the expression.

Why does step 2 work? A product is zero exactly when at least one of its factors is zero.

Simplifying

Just as \(\frac{18}{24} = \frac{3 \cdot 6}{4 \cdot 6} = \frac{3}{4}\), you can divide the top and bottom of a rational expression by a common factor.

Opposite factors

You'll often meet factors like \((a - b)\) and \((b - a)\). They aren't equal, but they're opposites:

So their quotient is \(-1\).

Practice