Algebraic Fractions

8 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team

An algebraic fraction has a polynomial on the top, the bottom or both, such as (x² − 9)/(x + 3). They follow exactly the same rules as ordinary fractions, and factoring is the key skill. If you are comfortable with <a href="/guides/how-to-add-and-subtract-fractions">numerical fractions</a> and <a href="/guides/factoring-polynomials">factoring</a>, this is the natural next step.

Excluded values

The denominator can never be zero, so some values of x are not allowed. In 1/(x − 2), x cannot be 2. Always note excluded values before you simplify, because they stay excluded even if a factor cancels.

Simplifying

Factor top and bottom and cancel common factors. (x² − 9)/(x + 3) = (x − 3)(x + 3)/(x + 3) = x − 3, for x ≠ −3. Cancel factors, not terms: you cannot cancel the x in (x + 3)/x.

Multiplying and dividing

Multiply: factor everything, cancel, then multiply what remains. Divide: flip the second fraction and multiply. Example: (x + 1)/(x − 2) × (x − 2)/(x + 5) = (x + 1)/(x + 5).

Adding and subtracting

Find a common denominator, just as with numbers. 1/x + 2/(x + 1) = (x + 1)/(x(x + 1)) + 2x/(x(x + 1)) = (3x + 1)/(x(x + 1)).

Solving equations with fractions

Multiply every term by the common denominator to clear the fractions. Solve 3/x + 1 = 4: multiply by x to get 3 + x = 4x, so x = 1. Check in the original: 3/1 + 1 = 4 ✓. Always reject any answer that makes a denominator zero.