Simplifying Rational Expressions

How to simplify rational expressions

The strategy: factor both numerator and denominator completely, then cancel common factors.

Step-by-step: Simplify $\dfrac{x^2 - 9}{x^2 + 5x + 6}$

Step 1 — Factor the numerator. $x^2 - 9$ is a difference of squares:

$x^2 - 9 = (x - 3)(x + 3)$

Step 2 — Factor the denominator. Find two numbers that multiply to 6 and add to 5: $2$ and $3$.

$x^2 + 5x + 6 = (x + 2)(x + 3)$

Step 3 — Write the factored form:

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

Step 4 — Cancel the common factor $(x + 3)$:

$\frac{x - 3}{x + 2}, \quad x \neq -3$

Important: Even though $(x + 3)$ cancels, $x = -3$ is still excluded from the domain — it makes the original denominator zero. The simplified form has a "hole" at $x = -3$ that doesn't show as an asymptote.

Step 5 — Check with a test value. Try $x = 1$: Original = $(1-9)/(1+5+6) = -8/12 = -2/3$. Simplified = $(1-3)/(1+2) = -2/3$ ✓.

Common mistakes

• Canceling terms instead of factors. $\frac{x^2 + 3}{x + 3} \neq x$. You can only cancel factors (things being multiplied), not terms (things being added).
• Forgetting domain restrictions. The cancelled factor still restricts the domain.

Try it in the visualization

Select from several expressions. Watch the factor-cancel animation — common factors highlight and disappear. The graphs of the original and simplified expressions are overlaid (identical except at the hole).