Algebraic fractions
To simplify algebraic fractions, factor first and cancel common factors, never separate terms. To add or subtract, use a common denominator; to divide, multiply by the reciprocal. Keep every restriction from the original expression.
What this guide covers
Multiplying, dividing, adding and subtracting algebraic fractions, including linear and quadratic denominators. Restrictions from the original denominator remain after simplification.
Syllabus reference: 2026 Mathematics 4052, N5, page 6. These are Mentora search groupings, not new official syllabus categories.
Four algebraic-fraction rules
- Simplify
- Factor the numerator and denominator, then cancel common factors. Keep excluded values.
- Multiply
- Multiply numerators and denominators; factor and cancel where possible.
- Add or subtract
- Use a common denominator. Subtract the whole second numerator, keeping its brackets.
- Divide
- Multiply by the reciprocal. The divisor must also be non-zero.
Worked example
Simplify (x² − 9)/(x² + 3x).
Factor the numerator as (x − 3)(x + 3) and the denominator as x(x + 3). Cancel the common factor x + 3 to obtain (x − 3)/x. The original expression excludes x = 0 and x = −3; both restrictions remain.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
Simplify 6a²/(9a), where a ≠ 0.
Show worked answer
Cancel the common factor 3a: 6a²/(9a) = 2a/3.
Check 2
Simplify 1/x + 2/(x + 1), where x ≠ 0, −1.
Show worked answer
The common denominator is x(x + 1). The numerator becomes (x + 1) + 2x = 3x + 1, giving (3x + 1)/(x(x + 1)).
Check 3
Simplify (x² − 4)/(x − 2), stating the restriction.
Show worked answer
Factor the numerator: (x − 2)(x + 2). Cancelling gives x + 2, but x ≠ 2 because the original denominator would be zero.
Common mistakes to watch for
- Do not cancel terms across addition: (x + 2)/x is not 2.
- When dividing by a tion, multiply by its reciprocal and exclude values that make the divisor zero or undefined.
- Keep brackets around the entire numerator when subtracting tions; the minus sign applies to every term.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
Related G3 Math guides
You may need first
These verified prerequisite links suggest useful foundations. They do not make a claim about your current readiness.
- Factorisation and subject of formulaFactor numerator and denominator to expose whole factors that can be cancelled.Practise factorisation →
- Find the Lowest Common Multiple (LCM) of two or more numbers by listing multiples or using prime factorisation. The LCM is the smallest number that is a multiple of all given numbers.Common denominators play the same role as an LCM: 1/6 + 1/4 becomes 2/12 + 3/12.Practise algebraic fractions →
Choose a useful next resource
Choose one step that matches the gap you just found.
More G3 Math topicsPublished by Mentora · Sources checked 11 Sept 2026
Editorial and source details
Original public examples and answers were checked with automated mathematical/editorial QA. No individual author, human reviewer or independent expert review is recorded.