Algebraic Fractions - Simplify, Add, Subtract, Multiply & Divide

What Are Algebraic Fractions?

An algebraic fraction (also called a rational expression) is a fraction where the numerator and/or denominator contain variables. Just like regular fractions, algebraic fractions follow the same rules - but with polynomials instead of numbers.

Examples: (x + 2) / 3, x² / (x - 1), (2x + 6) / (x + 3)

Key Rule: The denominator can never equal zero. For (x + 2)/(x - 3), x cannot be 3 because that would make the denominator 0. Always state restrictions.

How to Simplify Algebraic Fractions

Simplifying works just like regular fractions: factor the numerator and denominator, then cancel common factors.

Example: Simplify (x² - 4) / (x + 2)

Factor the numerator: x² - 4 = (x + 2)(x - 2)
Rewrite: (x + 2)(x - 2) / (x + 2)
Cancel (x + 2): (x - 2), where x ≠ -2

Try It: Simplify (x² - 9) / (x - 3).

Factor: (x + 3)(x - 3) / (x - 3)
Cancel (x - 3): x + 3, where x ≠ 3

Adding and Subtracting Algebraic Fractions

Just like regular fractions, you need a common denominator before adding or subtracting.

Same denominator:

(3x) / 5 + (2x) / 5 = 5x / 5 = x

Different denominators - find the LCD:

2/(x + 1) + 3/(x + 2)
LCD = (x + 1)(x + 2)
= 2(x + 2) / (x + 1)(x + 2) + 3(x + 1) / (x + 1)(x + 2)
= (2x + 4 + 3x + 3) / (x + 1)(x + 2)
= (5x + 7) / (x + 1)(x + 2)

Multiplying Algebraic Fractions

Multiply numerator × numerator and denominator × denominator, then simplify.

(x / 3) × (6 / x²) = 6x / 3x² = 2 / x
((x + 1) / x) × (x / (x + 2)) = x(x + 1) / x(x + 2) = (x + 1) / (x + 2)

Dividing Algebraic Fractions

To divide, flip the second fraction (take the reciprocal) and multiply.

(x / 4) / (x / 8) = (x / 4) × (8 / x) = 8x / 4x = 2

Quick Reference: Operations Summary

Operation	Rule	Example
Simplify	Factor, then cancel	(2x + 4)/(x + 2) = 2
Add/Subtract	Find LCD first	1/x + 1/(2x) = 3/(2x)
Multiply	Top × top, bottom × bottom	(x/2)(4/x) = 2
Divide	Flip second, then multiply	(x/3) / (x/6) = 2

Common Mistakes

Canceling terms instead of factors: (x + 3)/(x + 5) does NOT simplify. You can only cancel multiplied factors, not added terms.
Forgetting restrictions: Always note values that make the denominator zero.
Wrong LCD: When adding fractions, don't just multiply denominators - find the least common denominator for cleaner results.

Remember: Algebraic fractions follow the exact same rules as numerical fractions. If you can work with 2/3 and 5/6, you can work with x/(x+1) and 5/(x+2). Factor first, then cancel common factors.

Try our Factoring Calculator or Polynomial Calculator for more practice.