Algebraic Fractions

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Manipulate algebraic fractions
  2. Factorise and simplify rational expressions

What are Algebraic Fractions?

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An algebraic fraction is simply a fraction that contains letters (variables) as well as numbers. Just like a normal fraction has a top and bottom, an algebraic fraction has:

  • A numerator (the top part)
  • A denominator (the bottom part)

At least one of these parts contains a letter (like x, a, or p).

Examples of algebraic fractions:

  • x/3 (x on top, 3 on bottom)
  • (2x + 5)/(x - 1) (both top and bottom have algebra)
  • 3a/(4b) (letters on both top and bottom)

Algebraic fractions work in the same way as number fractions — you can simplify them, add them, subtract them, multiply them, and divide them. The rules are exactly the same, but you need to be careful with the algebra.

Simplifying Algebraic Fractions

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Simplifying means making a fraction as simple as possible by canceling out common factors.

Step-by-step method:

Step 1: Factorise the top and bottom fully

Look for common factors in both the numerator and denominator. Take out any numbers or letters that appear in all terms.

Step 2: Cancel common factors

If the same factor appears on both the top and bottom, you can cancel it out (divide both by that factor).

Examples:

Example 1: Simple factorising

Simplify: 2x/(x² + 3x)

  • First, factorise the bottom: x² + 3x = x(x + 3)
  • Now we have: 2x/(x(x + 3))
  • The factor x appears on both top and bottom, so cancel it
  • Answer: 2/(x + 3)

Example 2: Factorising with brackets

Simplify: (4x + 6)/(2x² - 7x - 15)

  • Factorise the top: 4x + 6 = 2(2x + 3)
  • Factorise the bottom: 2x² - 7x - 15 = (2x + 3)(x - 5)
  • Now we have: 2(2x + 3)/((2x + 3)(x - 5))
  • The bracket (2x + 3) appears on both top and bottom, so cancel it
  • Answer: 2/(x - 5)

Example 3: Simplifying with x terms

Simplify: (x² - 2x)/(x² - 5x + 6)

  • Factorise the top: x² - 2x = x(x - 2)
  • Factorise the bottom: x² - 5x + 6 = (x - 2)(x - 3)
  • Now we have: x(x - 2)/((x - 2)(x - 3))
  • Cancel (x - 2) from top and bottom
  • Answer: x/(x - 3)

Important warning:

You can ONLY cancel factors that are common to ALL terms on the top and ALL terms on the bottom.

Common mistake: You CANNOT simplify 6x/(x + 1) by canceling x, because x is not a factor of the entire bottom (it's only part of the bottom). The bottom does not factorise further.

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