Algebraic Fractions

What Is an Algebraic Fraction?

An algebraic fraction is a fraction in which the numerator, the denominator, or both contain variables. Just like numeric fractions such as 3/4, algebraic fractions such as (x + 2) / (x - 5) follow the same rules for simplifying, multiplying, dividing, adding, and subtracting — the difference is that the terms involve letters instead of only numbers.

Because the denominator of a fraction can never equal zero, algebraic fractions come with an important extra step: identifying any values of the variable that would make the denominator zero. Those values must be excluded from the fraction's domain. For example, in 5 / (x - 4), the value x = 4 is excluded because it would make the denominator zero.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, factor the numerator and denominator completely, then cancel any factors that appear in both.

Worked example: Simplify (x² - 9) / (x + 3).

• Factor the numerator using difference of squares: x² - 9 = (x + 3)(x - 3)
• The fraction becomes (x + 3)(x - 3) / (x + 3)
• Cancel the shared (x + 3) factor, leaving x - 3

Worked example: Simplify (x² + 5x + 6) / (x² - 4).

• Factor the numerator: x² + 5x + 6 = (x + 2)(x + 3)
• Factor the denominator: x² - 4 = (x + 2)(x - 2)
• Cancel the shared (x + 2) factor, leaving (x + 3) / (x - 2)

A common error is cancelling individual terms instead of factors. In (x + 3) / (x + 5), the x cannot be cancelled because it is added to other terms, not multiplied by the whole expression. Cancelling only works with factors that multiply the entire numerator and entire denominator.

Multiplying and Dividing

Multiplying algebraic fractions works the same way as multiplying numeric fractions: multiply the numerators together and multiply the denominators together, then simplify.

Worked example: Multiply (2x / 3) × (9 / 4x).

• Multiply straight across: (2x × 9) / (3 × 4x) = 18x / 12x
• Cancel the common factor of x: 18/12
• Simplify: 3/2

Dividing by an algebraic fraction means multiplying by its reciprocal (flip the second fraction upside down, then multiply).

Worked example: Divide (x / 2) ÷ (x / 6).

• Flip the second fraction and multiply: (x / 2) × (6 / x)
• Cancel x: 6/2
• Simplify: 3

Adding and Subtracting

Adding or subtracting algebraic fractions requires a common denominator, just like with numeric fractions. When the denominators are different, multiply each fraction so both share the same denominator before combining the numerators.

Worked example: Add 1/x + 2/(x + 1).

• The common denominator is x(x + 1)
• Rewrite each fraction: (x + 1) / [x(x + 1)] + 2x / [x(x + 1)]
• Add the numerators: (x + 1 + 2x) / [x(x + 1)]
• Simplify the numerator: (3x + 1) / [x(x + 1)]

Notice that x and (x + 1) cannot be cancelled here — they are separate factors that only combine once the fractions share the same denominator.

Common Mistakes

• Cancelling terms instead of factors — only fully factored expressions can be cancelled, and only when a factor appears in both the numerator and denominator.
• Forgetting excluded values — any value that makes the original denominator zero must be stated as excluded, even after simplifying.
• Adding denominators directly — like numeric fractions, algebraic fractions need a common denominator before the numerators can be added or subtracted.
• Skipping full factoring — a fraction may look simplified but still share a hidden common factor once both parts are properly factored.

Tips and Tricks

• Always factor first — simplifying, multiplying, and dividing algebraic fractions all start with factoring the numerator and denominator.
• Write out excluded values as soon as you see the original denominator, before any cancelling happens.
• Check your simplified answer makes sense by substituting a simple number (avoiding excluded values) into both the original and simplified fraction — the results should match.
• Treat "divide by a fraction" as "multiply by its reciprocal" every time; it removes a common source of errors.