Simplifying Rational Expressions

In this lesson you’ll learn what rational expressions are and how to simplify them by factoring the numerator and denominator completely.

A rational expression is a fraction where the numerator and/or denominator is a polynomial.

To simplify a rational expression:

1. Factor the numerator completely.
2. Factor the denominator completely.
3. Cancel any common factors (things that are multiplied in both).
4. Write the simplified expression in lowest terms.

Important rules:

• You can only cancel factors (things multiplied), never terms that are added or subtracted.
• Always note the values that make any denominator zero. These are excluded from the domain.

1. Simplify (x² − 9) / (x² − 6x + 9)

Factor both:

$$\frac{x^2 - 9}{x^2 - 6x + 9} = \frac{(x - 3)(x + 3)}{(x - 3)(x - 3)}$$

Cancel the common (x − 3):

$$= \frac{x + 3}{x - 3} \qquad (x \neq 3)$$

2. Simplify (6x² − 12x) / (3x² − 6x)

Factor both by pulling out the GCF:

$$\frac{6x(x - 2)}{3x(x - 2)} = \frac{6}{3} = 2 \qquad (x \neq 0,\; x \neq 2)$$

3. Simplify (x² + 5x + 6) / (x² − 4)

Factor both:

$$\frac{(x + 2)(x + 3)}{(x - 2)(x + 2)} = \frac{x + 3}{x - 2} \qquad (x \neq -2,\; x \neq 2)$$

Simplifying rational expressions is useful when:

• Finding average cost per unit (total cost / number of units)
• Simplifying rates in work problems
• Reducing formulas in physics or engineering
• Working with concentrations in mixtures

Example: if total cost is 2x² + 10x + 50 and you produce x items, average cost per item is (2x² + 10x + 50) / x. Simplifying helps analyze efficiency as production increases.