What is a rational expression?

A rational expression is a fraction whose numerator and denominator are polynomials. Examples include (x + 3)/(x - 2), (x² - 9)/(x + 3), and (2x² + 5x - 3)/(x² - 1). The denominator cannot be zero, so every rational expression has possible restrictions on the variable.

How do you simplify a rational expression?

First factor the numerator and denominator completely. Then cancel matching factors that appear as multiplication factors in both the numerator and denominator. Do not cancel pieces that are only terms inside a sum or difference. After cancelling, keep any excluded values from the original denominator.

Why do restrictions remain after a factor cancels?

Restrictions come from the original expression, not only from the simplified expression. If (x + 2) was in the original denominator, x = -2 made the original expression undefined. Cancelling the factor changes the appearance of the expression, but it does not make the original expression valid at that x-value.

Can I cancel x from x + 3 over x?

No. You can cancel common factors, not common terms. In (x + 3)/x, the numerator is a sum, so x is not a factor of the whole numerator. Cancelling it would change the value of the expression and give a wrong answer.

Why does the calculator sometimes say the expression is not supported?

The calculator focuses on the factoring patterns students usually meet first: linear factors and quadratics that factor over integers. If a quadratic has irrational or complex roots, or if the expression requires advanced symbolic manipulation, it is better to show a limitation than to give a fake simplified form.

Education

Rational Expression Simplifier Calculator

Simplify rational expressions the safe way: factor first, cancel only common factors, and keep the restrictions from the original denominator.

Enter a rational expression where the numerator and denominator are quadratic or linear expressions in x. The calculator factors what it can, cancels matching factors, and lists excluded x-values from the original denominator.

Numerator: ax² + bx + c

Denominator: dx² + ex + f

Simplified expression

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Original expression—

Factored form—

Restrictions—

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How to simplify rational expressions without losing restrictions

A rational expression is a fraction made from polynomials. The safe method is to factor the numerator and denominator first, then cancel only common factors. OpenStax explains this exact idea in its rational expressions lesson: cancellation is based on common factors, not terms that merely look similar.

The extra detail students often miss is the restriction. If a factor cancels, the original expression still had that factor in the denominator, so the value that made it zero is still excluded. This is why a simplified expression can have a “hole” even after it looks harmless.

Worked example: simplify (x² + 5x + 6) / (x² + 3x + 2)

x² + 5x + 6 = (x + 2)(x + 3)\nx² + 3x + 2 = (x + 1)(x + 2)\n\n(x + 2)(x + 3) / (x + 1)(x + 2) = (x + 3)/(x + 1)\n\nRestrictions from the original denominator: x ≠ -1, x ≠ -2

The cancelled factor x + 2 still matters because the original denominator was zero at x = -2.

What this calculator can and cannot simplify

This tool is designed for common classroom problems where the numerator and denominator factor into simple linear factors. It can handle many quadratic-over-quadratic examples. It does not pretend to be a full computer algebra system; if the expression needs advanced factoring, irrational roots, or several variables, it will tell you instead of returning a misleading answer.

Common questions

A rational expression is a fraction whose numerator and denominator are polynomials. Examples include (x + 3)/(x - 2), (x² - 9)/(x + 3), and (2x² + 5x - 3)/(x² - 1). The denominator cannot be zero, so every rational expression has possible restrictions on the variable.
First factor the numerator and denominator completely. Then cancel matching factors that appear as multiplication factors in both the numerator and denominator. Do not cancel pieces that are only terms inside a sum or difference. After cancelling, keep any excluded values from the original denominator.
Restrictions come from the original expression, not only from the simplified expression. If (x + 2) was in the original denominator, x = -2 made the original expression undefined. Cancelling the factor changes the appearance of the expression, but it does not make the original expression valid at that x-value.
No. You can cancel common factors, not common terms. In (x + 3)/x, the numerator is a sum, so x is not a factor of the whole numerator. Cancelling it would change the value of the expression and give a wrong answer.
The calculator focuses on the factoring patterns students usually meet first: linear factors and quadratics that factor over integers. If a quadratic has irrational or complex roots, or if the expression requires advanced symbolic manipulation, it is better to show a limitation than to give a fake simplified form.