Sheet № 143 · Higher only · AQA · Edexcel · OCR
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
§Key definitions
Question:
Simplify (6x²)/(9x).
Answer:
(x − 3)/(x + 2)
Q1 (Foundation):
Simplify (4x³)/(10x).
Q2 (Higher):
Simplify (x² − 4)/(x² − 4x + 4).
Q3 (Higher):
Simplify (3x² + 7x + 2)/(3x² − 5x − 2).
§Formulas to memorise
Factorise the numerator and denominator, then cancel common factors
You can only cancel factors (terms that multiply), never individual terms that are added or subtracted
Factorise the numerator — completely.
Factorise the denominator — completely.
Identify common factors — that appear in both.
Cancel the common factors — by dividing them out.
Write the simplified fraction. — If everything cancels from the numerator, you are left with 1 (not 0).
Worked example
Simplify (6x²)/(9x).
Working:
⚠ 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.