Simplifying Rational Expressions
Factor first, cancel factors only, and record what x cannot be.
The explanation
A rational expression is a fraction with polynomials on top and bottom. Simplify it exactly like a numeric fraction: factor both parts, then cancel common factors.
(x² − 9)/(x² + 7x + 12) factors to ((x+3)(x−3))/((x+3)(x+4)), and the (x+3) cancels, leaving (x − 3)/(x + 4).
You may only cancel *factors* — things being multiplied. You can never cancel across addition. In (x + 5)/x, the x does not cancel.
State the restrictions. The original was undefined at x = −3 and x = −4, and both stay excluded even though −3 vanished from the simplified form.
A rational expression P(x)/Q(x) is defined wherever Q(x) ≠ 0. Simplifying divides numerator and denominator by a common factor, which is multiplication by 1 only on the set where that factor is nonzero.
This is why domain restrictions come from the *original* expression, not the simplified one. Cancelling (x + 3) from ((x+3)(x−3))/((x+3)(x+4)) produces a function that is defined at x = −3 while the original is not; graphically the original has a removable discontinuity, a hole, at that point, and the simplified form does not.
Cancelling across addition is invalid because addition is not a factorisation — (x + 5)/x has no common factor. This error is the algebraic equivalent of writing (3 + 4)/3 = 4.
A useful special case: (a − b) and (b − a) differ by a factor of −1, so (x − 5)/(5 − x) = −1 for x ≠ 5. Factoring out −1 makes such cancellations legitimate and visible.
Worked example
Simplify (2x² − 8)/(x² − x − 2) and state the restrictions.
- Numerator: 2(x² − 4) = 2(x + 2)(x − 2).
- Denominator: (x − 2)(x + 1).
- Cancel (x − 2): 2(x + 2)/(x + 1).
- Original denominator is zero at x = 2 and x = −1.
Answer: 2(x + 2)/(x + 1), with x ≠ 2 and x ≠ −1 (a hole at x = 2).
Common mistakes
- Cancelling individual terms rather than whole factors.
- Dropping the restriction that came from a cancelled factor.