Rational Expressions

Rational expressions are algebraic fractions where both the numerator and denominator are polynomials. The denominator cannot be zero, so every rational expression has a domain.

The above expression is defined if x ≠ 3, because then the denominator would be zero.

Domain

The domain is the set of x values for which the denominator is not zero. This must always be determined separately.

• If the denominator is x - 5, then x ≠ 5.
• If the denominator is x² - 4, then x ≠ ±2.
• If the denominator is (x + 1)(x - 7), then x ≠ -1 and x ≠ 7.

Simplification

We simplify rational expressions by factoring the numerator and denominator, then canceling common factors. It is important to still indicate the excluded values.

Operations with Rational Expressions

• Addition: bring to a common denominator then add the numerators.
• Subtraction: bring to a common denominator then subtract the numerators.
• Multiplication: multiply numerators together, denominators together.
• Division: multiply by the reciprocal of the second fraction.

Example of multiplication:

Addition Example

Rational Expressions in Practice

Rational expressions often appear in physics, economics, and engineering calculations. For example:

• Speed = distance / time → if distance and time are described by polynomials, the result is a rational expression.
• When calculating average speed, the denominator is always the total time, which cannot be zero.
• In chemical reaction rate laws, rational ratios often appear.

Practice Exercise