Rational Expressions • Topic 1 of 6

Simplification

To simplify a rational expression, factor the numerator and denominator completely, then cancel any common factors. So (x² − 9)/(x − 3) = (x − 3)(x + 3)/(x − 3) = x + 3. You can only cancel factors (things multiplied), never individual terms across a + or −. State restrictions: the cancelled value (here x ≠ 3) is still excluded from the domain. Recognising the difference of squares and simple trinomial factorings is the key skill. Simplifying first makes the four operations and equation-solving far cleaner, and the SAT rewards spotting the factorisation quickly.

✅ Solved examples

1. Simplify (x² − 9)/(x − 3).
Factor: (x − 3)(x + 3)/(x − 3) = x + 3 (x ≠ 3).
2. Simplify (x² + 5x + 6)/(x + 2).
(x + 2)(x + 3)/(x + 2) = x + 3.
3. Simplify (4x + 8)/(x + 2).
4(x + 2)/(x + 2) = 4.
4. Simplify 6x²/(2x).
6x²/(2x) = 3x (x ≠ 0).

✏️ Practice — try these, take hints as needed

1. Simplify (x² − 25)/(x − 5).
Factor the numerator (difference of squares).
(x − 5)(x + 5).
Cancel (x − 5).
x + 5.
2. Simplify (x² + 7x + 12)/(x + 3).
Factor the trinomial.
(x + 3)(x + 4).
Cancel (x + 3).
x + 4.
3. Simplify (3x + 9)/(x + 3).
Factor out 3.
3(x + 3)/(x + 3).
—
3.
4. Simplify 10x³/(5x).
Divide coefficients and subtract exponents.
10/5 = 2, x³/x = x².
—
2x².
5. Simplify (x² − x − 6)/(x − 3).
Factor: (x − 3)(x + 2).
Cancel (x − 3).
—
x + 2.

📝 Topic test — 8 questions

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