Algebraic fractions follow the same rules as number fractions. To simplify, factorise the top and bottom and cancel common factors, never individual terms. To multiply, multiply the tops and the bottoms; to divide, flip the second fraction and multiply. To add or subtract, use a common denominator. Equations with algebraic fractions are solved by multiplying through by the denominators.
Algebraic fraction rules
- Simplify: factorise, then cancel common factors only
- Multiply: tops × tops, bottoms × bottoms
- Divide: flip the second fraction, then multiply
- Add or subtract: common denominator, then combine the numerators
- Equations: multiply every term by the common denominator
Simplifying algebraic fractions
Factorise the numerator and the denominator fully, then cancel any bracket or factor that appears in both. (x² − 9)/(x² + 5x + 6) factorises to (x − 3)(x + 3)/((x + 2)(x + 3)), and cancelling (x + 3) leaves (x − 3)/(x + 2). You can cancel factors, which are multiplied, but never terms, which are added.
Multiplying and dividing
To multiply, factorise everything first, cancel common factors across the fractions, then multiply what is left. To divide, flip the second fraction and multiply. For example, 2x/(x + 4) ÷ 6/(x + 4) = 2x/(x + 4) × (x + 4)/6 = 2x/6 = x/3.
Adding and subtracting
Use a common denominator, as with number fractions. For 3/(x + 1) + 2/(x − 2), the common denominator is (x + 1)(x − 2). The numerators become 3(x − 2) and 2(x + 1), which add to 5x − 4. When subtracting, put the second numerator in brackets so the minus applies to all of it.
Solving equations with algebraic fractions
Multiply every term by the common denominator to clear the fractions. For 3/(x − 1) − 2/(x + 1) = 1, multiplying by (x − 1)(x + 1) gives 3(x + 1) − 2(x − 1) = (x − 1)(x + 1). This simplifies to a quadratic, x² − x − 6 = 0, which you can then solve.
Checking answers
Substitute your solution back into the original equation, and make sure it does not make any denominator zero. A value that gives a zero denominator must be rejected.
Worked examples
Example 1. Simplify (x² − 9)/(x² + 5x + 6).
- Factorise: (x − 3)(x + 3) on top and (x + 2)(x + 3) on the bottom.
- Cancel the common factor (x + 3).
Answer: (x − 3)/(x + 2).
Example 2. Simplify (x² − 4)/(3x) × 6x/(x + 2).
- Factorise: (x − 2)(x + 2)/(3x) × 6x/(x + 2).
- Cancel (x + 2), and 6x ÷ 3x = 2.
Answer: 2(x − 2).
Example 3. Simplify 2x/(x + 4) ÷ 6/(x + 4).
- Flip and multiply: 2x/(x + 4) × (x + 4)/6.
- Cancel (x + 4): 2x/6.
Answer: x/3.
Example 4. Write 3/(x + 1) + 2/(x − 2) as a single fraction.
- Common denominator (x + 1)(x − 2).
- Numerator: 3(x − 2) + 2(x + 1) = 5x − 4.
Answer: (5x − 4)/((x + 1)(x − 2)).
Example 5. Solve 3/(x − 1) − 2/(x + 1) = 1.
- Multiply by (x − 1)(x + 1): 3(x + 1) − 2(x − 1) = x² − 1.
- x + 5 = x² − 1, so x² − x − 6 = 0.
- (x − 3)(x + 2) = 0.
Answer: The solutions are x = −2 and x = 3.
Common mistakes
- Cancelling terms instead of factors. (x + 3)/(x + 5) does not simplify to 3/5.
- Not factorising first. Factorise both the top and the bottom before cancelling.
- Sign errors when subtracting. Put the second numerator in brackets before expanding.
- Multiplying only some terms. When clearing fractions in an equation, multiply every term.
- Keeping a solution that gives a zero denominator. Check every solution in the original equation.
How algebraic fractions are examined
Algebraic fractions are part of the Higher tier for both Edexcel GCSE (1MA1) and the International GCSE (4MA1). Questions often carry 3 to 5 marks and ask you to:
- simplify a fraction by factorising a quadratic
- add or subtract two algebraic fractions
- multiply or divide algebraic fractions
- solve an equation that contains algebraic fractions, leading to a quadratic
The algebraic fractions worksheet includes worked answers. For extra questions, use the algebra engine.
This needs confident factorising and fraction arithmetic. At A-Level, see algebraic fractions.
Frequently asked questions
How do you simplify an algebraic fraction?
Factorise the numerator and denominator, then cancel any factors they share.
How do you add algebraic fractions?
Write them over a common denominator, then add the numerators.
Can you cancel the x in (x + 2)/(x + 7)?
No. x is a term, not a factor, so nothing cancels.