Method guideCalculus I · Algebra IIIntermediate

Simplifying rational expressions by factors, not by terms

Factor completely, record excluded inputs, and cancel common factors. Individual terms separated by addition cannot be canceled.

LaTeX article Updated July 13, 2026

\frac{ab}{ac}=\frac bc,\quad a\ne0

Terms are not factors

In (x + 3)/x, the x in the numerator is one term of a sum, not a factor multiplying the entire numerator. It cannot cancel with the denominator.

Factor bars, not visual symbols. A legal cancellation should be explainable as dividing the full numerator and denominator by the same nonzero expression.

\frac{x+3}{x}\ne3

Factor every polynomial first

Use GCF, trinomial, and special-pattern factoring until numerator and denominator are products. Then identify identical factors.

Signs matter: x − 4 and 4 − x are opposites, so one can be rewritten as −(x − 4).

4-x=-(x-4)

Keep the domain attached

The simplified expression and original expression agree only on the original domain. State the canceled restrictions beside the answer.

This makes later graphing and equation solving honest because holes do not disappear from the mathematics.

\frac{x^2-16}{x-4}=x+4,\quad x\ne4

Worked example

Common mistakes

Canceling terms across addition.
Factoring only the denominator.
Dropping a restriction when its factor cancels.

Keep these ideas

Factor before canceling.
Only common factors cancel.
Carry the original domain.