Algebraic Fractions Calculator with Steps

What makes an algebraic fraction different?

An algebraic fraction has a variable in its numerator or denominator. Its value is defined only when every original denominator is nonzero. Before simplifying (x^2 − 1)/(x − 1), note that x ≠ 1. Factoring the numerator gives (x − 1)(x + 1). Cancelling x − 1 leaves x + 1, with x = 1 still excluded.

That exclusion describes the domain of the original expression. Although x + 1 can be evaluated at x = 1 on its own, the original fraction cannot. The calculator reports both the simplified form and the restrictions so the result is not mistaken for a new unrestricted expression.

How the fraction operations work

• Simplify: factor the numerator and denominator, cancel their common factor, and reduce numeric coefficients. Cancel factors, never separate terms in a sum.
• Add or subtract: use a common denominator, combine the complete numerators, and then simplify. For 1/x + 1/(x + 1), the numerator becomes (x + 1) + x = 2x + 1, with denominator x(x + 1).
• Multiply: multiply numerators and denominators. Common factors may be cancelled, but original exclusions remain.
• Divide: multiply by the reciprocal of the second fraction. The second fraction must be defined and nonzero, so its original numerator adds another restriction.

For (x + 1)/(x − 1) ÷ (x + 1)/(x + 2), the simplified result is (x + 2)/(x − 1). However, x = 1 and x = −2 are excluded by the original denominators, and x = −1 is excluded because it makes the divisor zero.

Avoid the most common cancellation error

(x + 2)/x does not simplify to 2. The numerator is a sum, so x is not a factor of the whole numerator. You can rewrite it as 1 + 2/x for x ≠ 0, but cannot cancel an x from just one term and discard the remaining denominator.

In contrast, (2x + 4)/2 does simplify to x + 2 because the constant 2 divides every numerator term. The whole numerator has a common factor 2. Writing brackets around the numerator and denominator helps preserve this distinction.

Supported scope and interpreting the domain

This calculator accepts polynomial numerator and denominator fields in x of degree 2 or less. Combining two inputs may produce a higher-degree intermediate expression; the common polynomial factors are cancelled exactly. Numeric coefficients and constant fractions are supported.

A zero denominator is rejected. A zero numerator is allowed and simplifies to zero, but the original exclusions remain. Dividing by an identically zero fraction is rejected. Excluded real roots are listed when an original denominator or divisor numerator is linear or quadratic. A quadratic with no real roots adds no real exclusion.

Rational expressions with several variables, nested variable fractions, radicals or functions are outside this tool. Use the numeric fraction calculator for mixed-number arithmetic, or the expression calculator for polynomial expansion.

Further reading: OpenStax: polynomials and rational expressions.