Simplification

What’s a Rational Expression?

A fraction where the numerator and denominator are polynomials.

$\frac{x + 1}{x - 2} \qquad \frac{x^2 - 4}{x + 2} \qquad \frac{2x}{x^2 + 1}$

Simplifying Rational Expressions

Same idea as simplifying $\frac{6}{8}$ to $\frac{3}{4}$: cancel common factors.

Example: Simplify $\dfrac{x^2 - 4}{x + 2}$

Step 1: Factor

The numerator is a difference of squares:

$x^2 - 4 = (x + 2)(x - 2)$

Step 2: Cancel common factors

$\frac{(x + 2)(x - 2)}{x + 2} = x - 2$

The $(x + 2)$ cancels.

The Key Rule

You can only cancel factors, not terms.

Wrong:

$\frac{x + 1}{x + 2} \neq \frac{1}{2}$

You cannot cancel the $x$ because it’s a term, not a factor.

Right:

$\frac{x(x + 1)}{x(x + 2)} = \frac{x + 1}{x + 2}$

Here $x$ is a factor of both numerator and denominator.

More Examples

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

$\frac{x^2 + 5x + 6}{x + 2} = \frac{(x+2)(x+3)}{x+2} = x + 3$

$\frac{2x^2 + 4x}{2x} = \frac{2x(x + 2)}{2x} = x + 2$