Simplify Rational Expressions Calculator
The answer to a question like this has two parts, and almost everyone writes one of them.
Simplify (x² − 4) / (x² + 5x + 6)
(x − 2) / (x + 3), x ≠ −2, x ≠ −3
Both halves are the answer. Leave the restrictions off and the work is not finished, which is why this simplify rational expressions calculator puts them beside the result rather than buried in the working.
Simplifying Rational Expressions Starts With Factoring
Nothing can be cancelled until both parts are products:
(x² − 4) / (x² + 5x + 6) = (x − 2)(x + 2) / (x + 2)(x + 3)
Now the shared factor is visible. Before factoring it was not, and looking for something to cancel in the unfactored form is how people end up cancelling an x out of x² − 4 — which is not allowed, because that x is part of a sum, not a factor.
If the factoring itself is the sticking point, the factoring polynomials calculator does that half on its own.
Write the Restrictions Before You Cancel
This is the step that decides whether the answer is right.
The bottom is (x + 2)(x + 3), so the expression is undefined at x = −2 and x = −3. Write both down now, while both factors are still there.
Then cancel (x + 2). The answer is (x − 2)/(x + 3) — and x = −2 is still excluded, even though nothing in that answer says so. The original expression had no value at −2, and rewriting an expression cannot give it one.
Take the restrictions from the simplified form instead and you would only find x ≠ −3. That is the error, and it is worth one mark on nearly every question of this type.
Only Factors Cancel
Cancelling means dividing the top and bottom by the same thing, and only a whole factor can be divided out.
| (x + 2)(x − 2) / (x + 2)(x + 3) | ✓ (x + 2) is a factor of both |
| (x + 3) / (x + 5) | ✗ those x's are terms, nothing cancels |
The second one is already in lowest terms. It looks like something should cancel, and nothing does.
When Everything Cancels
(x² − 9)/(x² − 9) simplifies to 1 — but only where it is defined. The restrictions x ≠ 3 and x ≠ −3 still hold, so the answer is 1 everywhere except at those two points.
It is a strange-looking result and exam papers use it precisely because it separates people who understood the rule from people who followed the procedure.
How to Use This Simplify Rational Expressions Calculator
Put the numerator in the top box and the denominator in the bottom one, then press =. Brackets are expanded first, so (x−2)(x+2) works as well as x²−4.
The band gives the simplified expression with the restrictions underneath, and tells you when some of them are no longer visible in the answer. Tap the button to see the factoring, with the cancelled factor struck through, and the steps run through it more slowly below.
Frequently Asked Questions
How do you simplify a rational expression?
What are the restrictions on a rational expression?
Do restrictions disappear when a factor cancels?
Why can you only cancel factors, not terms?
Is this simplify rational expressions calculator free?
Three habits cover this whole topic: factor before you look for anything to cancel, write the restrictions down while the original denominator is still in front of you, and cancel only whole factors. The third is easy. The second is the one that pays.