Algebraic fractions
An algebraic fraction follows the same arithmetic rules as a numerical fraction. Factor before cancelling, and cancel whole common factors only. For addition, rewrite both fractions over a common denominator before combining numerators.
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.
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.
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.
Useful foundations to check
These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.
Factorisation and subject of formula
Factor numerator and denominator to expose whole factors that can be cancelled.
Practise factorisationFind 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
Applications and next topics
Use these operations when rearranging formulae and solving fractional equations. Check excluded values before accepting any solution.
Choose a useful next resource
To understand a recurring error rather than relearn the whole skill: Understand cancellation and sign errors.
Sources and review
Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-11; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.