How do I simplify a rational expression without changing its restrictions?
Quick answer
Factor the numerator and denominator, but first find the values that make the original denominator zero and write them down as restrictions. Cancel only common factors, not individual terms. Keep the original restrictions beside the simplified expression: cancellation can make the reduced formula defined at a value where the original expression was undefined, so their domains may differ. [S1]
The question, the details, the next step.
What order should I use so I don’t lose a restriction while simplifying?
Use this order: identify the original denominator, find the values that make it zero, factor the numerator and denominator, and then cancel common factors. Record the exclusions before canceling anything. A rational expression is undefined wherever its denominator is zero, so its restrictions come from the denominator as originally written. [S1]
For example, suppose the expression is (x² − x − 2)/(x² + x − 6). Before simplifying, factor the denominator: x² + x − 6 = (x + 3)(x − 2). It is zero at x = −3 and x = 2, so write x ≠ −3, 2 as the original restrictions. Those values must remain excluded even if a factor later cancels.
Can you show how the cancellation works in that example?
Factor the numerator too: x² − x − 2 = (x − 2)(x + 1). Then show the factored expression and cancel the shared factor:
[(x − 2)(x + 1)] / [(x + 3)(x − 2)] = (x + 1)/(x + 3)
The cancellation is valid for values in the original domain; it does not authorize using x = 2. At x = 2, the original denominator is zero, so the original expression is undefined. The original restrictions remain x ≠ −3, 2. The reduced formula by itself has denominator zero only at x = −3, but that does not restore x = 2 to the original expression’s domain. [S1]
So are the original expression and the reduced one equivalent, or are they different?
They give the same value wherever the original expression is defined, but they do not have identical domains if you look only at the reduced formula. In the example, (x + 1)/(x + 3) can be evaluated at x = 2, while the original expression cannot. Think of the answer as the simplified formula together with its inherited restrictions: (x + 1)/(x + 3), where x ≠ −3, 2. The restriction rule explains why canceled values still need to be excluded. [S1]
A useful habit is to keep two separate notes: “original restrictions” and “denominator of reduced formula.” The first describes the expression you were given; the second tells you where the written reduced formula itself is undefined.
What should I write on my homework so the restriction is clear?
Show the factoring and cancellation, then state the restrictions from the original denominator in the final line. For the example, a clear answer is:
(x² − x − 2)/(x² + x − 6)
= [(x − 2)(x + 1)]/[(x + 3)(x − 2)]
= (x + 1)/(x + 3), where x ≠ −3, 2
That final phrase makes clear you simplified the expression without silently expanding its domain. If a teacher or assignment uses a particular notation, follow that format while preserving the same exclusions.
A practical checklist
Copy the original denominator before changing the expression.
Solve for every value that makes that denominator zero and record those restrictions.
Factor the numerator and denominator completely where possible.
Cancel shared factors only; do not cancel terms separated by addition or subtraction.
Write the simplified expression together with all original restrictions.
A few more useful answers
If a factor cancels, can I include its zero in the answer’s domain?
Not for the original expression. Keep the value excluded if it made the original denominator zero, even when the reduced formula is defined there.
Do I find restrictions from the numerator too?
For this domain check, find restrictions by setting the original denominator equal to zero. A zero of the numerator alone does not make a rational expression undefined.
What if the denominator is already factored?
Use its factors to identify the values that make it zero, record those exclusions, and then proceed with simplification.
Sources
References used for this guide.
- [S1] OpenStax Intermediate Algebra 2e: Rational-Expression Domains openstax.org · Checked