Algebraic fractions follow exactly the same rules as number fractions, but you must factorise first. To simplify, factorise the top and bottom and cancel common factors. To multiply or divide, factorise everything and cancel before multiplying. To add or subtract, find a common denominator. The same skills tidy up quotient-rule answers and set up partial fractions.
The rules for algebraic fractions
- Simplify: factorise top and bottom, then cancel common factors
- Multiply: a/b × c/d = ac/bd, cancelling first
- Divide: a/b ÷ c/d = a/b × d/c
- Add or subtract: use a common denominator, built from the factorised denominators
You may only cancel factors (things multiplied), never terms (things added or subtracted).
How do you simplify an algebraic fraction?
Factorise the numerator and the denominator fully, then cancel any bracket that appears in both. The key rule is that you can only cancel factors, never terms. In (x + 3)/(x + 5) the x's are terms, so nothing cancels. Once both parts are written as brackets multiplied together, a common bracket is a factor and can be cancelled.
Look for a difference of two squares, such as x² − 9 = (x − 3)(x + 3), and for common factors to take out first. If nothing factorises, the fraction is probably already in its simplest form.
Multiplying and dividing algebraic fractions
Factorise every numerator and denominator before doing anything else, then cancel across the fractions: a factor on the top of one can cancel a factor on the bottom of the other. To divide, flip the second fraction and multiply.
Do not multiply out the brackets first. Factorised form is what lets you cancel, and expanding destroys it.
Adding and subtracting algebraic fractions
You need a common denominator. Factorise the denominators first, so you use the lowest common denominator rather than an unnecessarily large product. Rewrite each fraction over that denominator, then combine the numerators.
When subtracting, the minus sign applies to the whole of the second numerator. Put each subtracted numerator in brackets: (2x + 1) − (x − 2) = x + 3, not x − 1. This is the most common error in the topic.
Fractions within fractions
A fraction whose denominator contains another fraction collapses if you work from the inside out. First combine everything inside the denominator into a single fraction, writing 1 as 2/2 or similar where needed. Then divide by multiplying by the reciprocal, and cancel.
Do not expand a squared bracket before simplifying inside it. Simplifying first usually turns it into something very small.
Tidying quotient-rule answers
Algebraic fractions come back in differentiation. The quotient rule gives answers that look complicated until you simplify them, and examiners regularly report marks lost on the algebra rather than the calculus. Expand and collect the numerator, factorise it if you can, and leave the denominator in its factorised squared form so you can see whether anything cancels.
Solving equations with algebraic fractions
Multiply every term by the common denominator to clear the fractions, then solve the resulting equation. Before you start, write down any values of x that would make a denominator zero. If one of those appears as a solution, reject it, because multiplying through can create roots that do not satisfy the original equation.
Worked examples
Example 1. Simplify (x² − 9) / (x² + 7x + 12).
- Numerator, a difference of two squares: x² − 9 = (x − 3)(x + 3).
- Denominator: x² + 7x + 12 = (x + 3)(x + 4).
- Cancel the common factor (x + 3): (x − 3)/(x + 4).
- Check at x = 1: (1 − 9)/(1 + 7 + 12) = −0.4, and (1 − 3)/(1 + 4) = −0.4.
Answer: (x − 3)/(x + 4).
Marks: M1 for factorising both, A1 for cancelling.
Example 2. Simplify (x + 2)/(x − 1) × (x² − 1)/(x² − 4).
- Factorise: x² − 1 = (x − 1)(x + 1) and x² − 4 = (x − 2)(x + 2).
- Cancel (x + 2) across the fractions, and (x − 1) across the fractions.
- What remains is (x + 1)/(x − 2).
- Check at x = 3: (5/2) × (8/5) = 4, and 4/1 = 4.
Answer: (x + 1)/(x − 2).
Marks: M1 for factorising, M1 for cancelling across, A1 for the result.
Example 3. Write 2/(x + 1) + 3/(x − 2) as a single fraction in its simplest form.
- The denominators share no factor, so the common denominator is (x + 1)(x − 2).
- Rewrite: [2(x − 2) + 3(x + 1)] / [(x + 1)(x − 2)].
- Expand and collect the numerator: 2x − 4 + 3x + 3 = 5x − 1.
Answer: (5x − 1) / ((x + 1)(x − 2)).
Marks: M1 for the common denominator, M1 for combining, A1 for simplifying.
Example 4. Solve 1/x + 1/(x + 1) = 5/6.
- Excluded values: x ≠ 0 and x ≠ −1.
- Multiply every term by 6x(x + 1): 6(x + 1) + 6x = 5x(x + 1).
- Expand and collect: 5x² − 7x − 6 = 0.
- Factorise: (5x + 3)(x − 2) = 0, so x = −3/5 or x = 2. Neither is excluded.
- Check x = 2: ½ + ⅓ = 5/6.
Answer: x = −3/5 or x = 2.
Marks: M1 for clearing fractions, M1 for the quadratic, A1 for both roots.
Common mistakes
- Cancelling terms instead of factors. (x + 3)/(x + 5) does not simplify to 3/5. Factorise first, then cancel whole brackets.
- Dropping the bracket when subtracting. Write −(x − 2), not −x − 2. The minus sign applies to every term of the numerator.
- Expanding before cancelling. Keep expressions factorised while you multiply or divide; cancelling is much easier.
- Using a larger denominator than needed. Factorise the denominators first to find the lowest common denominator.
- Keeping an excluded value as a solution. A value that makes a denominator zero must be rejected.
How algebraic fractions are examined
In Edexcel International A-Level, algebraic fractions are Pure 3 content (paper WMA13), in the algebraic methods chapter. In Edexcel A-Level (9MA0) they are second-year Pure. Questions usually ask you to:
- simplify a fraction fully, often after factorising a cubic or quadratic
- express a sum or difference as a single fraction in its simplest form
- simplify a fraction within a fraction
- tidy an expression produced by the quotient rule
Try the Pure 3 algebraic methods exam questions with full mark schemes, or the algebraic methods worksheet. The A-Level algebra revision engine generates unlimited extra questions.
Next steps: improper algebraic fractions and then partial fractions, which reverses adding fractions.
Frequently asked questions
Can you cancel x in (x + 2)/(x + 5)?
No. The x's are terms, not factors. You can only cancel a factor that multiplies the whole numerator and the whole denominator.
How do you find the common denominator of algebraic fractions?
Factorise each denominator, then take every different factor the highest number of times it appears. If the denominators share no factor, their product is the common denominator.
Why do I need to check excluded values?
Multiplying through by the denominator can introduce solutions that make the original fractions undefined. Any such value must be rejected.