Simplifying Algebraic Fractions

Simplifying algebraic fractions is a Higher-only skill that builds directly on your factorising knowledge. The process is the same as simplifying numerical fractions — find common factors in the numerator and denominator, then cancel. The difference is that you need to factorise algebraic expressions first. This topic is tested regularly on AQA, Edexcel, and OCR Higher papers.

What Is Simplifying Algebraic Fractions?

An algebraic fraction has algebraic expressions in the numerator, the denominator, or both. Simplifying means reducing the fraction to its simplest form by cancelling common factors — just as you would simplify 12/18 to 2/3 by dividing both by 6.

Key Formulas

Step-by-Step Method

1. Factorise the numerator completely.
2. Factorise the denominator completely.
3. Identify common factors that appear in both.
4. Cancel the common factors by dividing them out.
5. Write the simplified fraction. If everything cancels from the numerator, you are left with 1 (not 0).

Worked Example 1 — Foundation Level

Question: Simplify (6x²)/(9x).

Working:

• Numerator: 6x² = 6 × x × x.
• Denominator: 9x = 9 × x.
• Common factors: 3 and x.
• Cancel: (6x²)/(9x) = (2x)/3.

Answer: 2x/3

Worked Example 2 — Higher Level

Question: Simplify (x² − 9)/(x² + 5x + 6).

Working:

• Factorise numerator: x² − 9 = (x + 3)(x − 3) (difference of two squares).
• Factorise denominator: x² + 5x + 6 = (x + 3)(x + 2) (two numbers that multiply to 6 and add to 5).
• Common factor: (x + 3).
• Cancel: (x + 3)(x − 3) / [(x + 3)(x + 2)] = (x − 3)/(x + 2).

Answer: (x − 3)/(x + 2)

Worked Example 3 — Exam Style

Question: Simplify (2x² + 5x − 3)/(2x² − x).

Working:

• Factorise numerator: 2x² + 5x − 3. Using ac method: ac = −6, numbers 6 and −1. Split: 2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (x + 3)(2x − 1).
• Factorise denominator: 2x² − x = x(2x − 1).
• Common factor: (2x − 1).
• Cancel: (x + 3)(2x − 1) / [x(2x − 1)] = (x + 3)/x.

Answer: (x + 3)/x

Common Mistakes

• Cancelling terms instead of factors. You cannot cancel the x² from (x² + 3)/(x² + 5). These are terms, not factors. You can only cancel factors that multiply the entire numerator and denominator.
• Not factorising fully before cancelling. If you do not factorise completely, you will miss common factors and leave the fraction unsimplified.
• Forgetting to state restrictions. Strictly, values that make the original denominator zero are excluded, though GCSE exams rarely require you to state this.

Exam Tips

• Factorise both parts fully before attempting to cancel anything.
• Look for difference of two squares, common factors, and quadratic factorisation.
• If the numerator or denominator has a common numerical factor, take it out first.
• After cancelling, check by substituting a value (e.g., x = 2) into both the original and simplified fraction — they should give the same result.

Practice Questions

Q1 (Foundation): Simplify (4x³)/(10x).

Q2 (Higher): Simplify (x² − 4)/(x² − 4x + 4).

Q3 (Higher): Simplify (3x² + 7x + 2)/(3x² − 5x − 2).

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Related Topics

• Algebraic Fractions
• Factorising Expressions
• Difference of Two Squares

Summary

• Simplifying algebraic fractions requires factorising the numerator and denominator first.
• Cancel common factors, not individual terms.
• Look for difference of two squares, single bracket, and quadratic factorisations.
• Always check your answer by substituting a value into both the original and simplified versions.
• This skill is a building block for adding algebraic fractions and solving equations with fractions.