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.
How to recognize the method, run it, and know when it is the wrong choice.
Updated July 13, 2026Cancellation is division by a common factor. If the expression is not factored, the shared object may not be visible—or may not exist.
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.
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).
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.
Simplify (x² + 5x + 6)/(x² − 9).
Factor the numerator.
Factor the denominator and note x ≠ ±3.
Cancel the common factor x + 3.
Common mistakes
- Canceling terms across addition.
- Factoring only the denominator.
- Dropping a restriction when its factor cancels.
Three takeaways
- Factor before canceling.
- Only common factors cancel.
- Carry the original domain.