Rational Expressions

Algebraic fractions: reduce, multiply, divide, add, and subtract.

The idea

A rational expression is a fraction whose numerator and denominator are both polynomials:

$\frac{p(x)}{q(x)}, \qquad q(x) \neq 0.$

The name echoes the rational numbers, one integer over another, and the rules for ordinary fractions carry over.

To reduce, factor the numerator and the denominator and cancel any factor they share, just as $\tfrac{6}{9}$ reduces to $\tfrac{2}{3}$. Factoring comes first because cancelling is legal only for a whole factor of the top and the bottom — never for one term inside a sum.

To multiply, multiply numerator by numerator and denominator by denominator, then cancel what the product shares.

To add or subtract, first rewrite both fractions over a common denominator; then combine the numerators and keep the denominator.

One point has no counterpart with numbers. The denominator contains a variable, so the values of $x$ that make it zero are excluded from the start — and they stay excluded even after a cancellation removes the factor that produced them.

Ways to work on it

• Walkthrough. Reduce, multiply, and add rational expressions.
• Practice. Reduce or multiply a single rational expression.
• Hardest. Add fractions with different denominators.

Not sure where to start? Take the ten-question placement test.