An algebraic fraction has an algebraic expression in the numerator, denominator, or both — for example, 3x/6 or (x² − 1)/(x + 1). The rules for simplifying, adding, subtracting, multiplying, and dividing them mirror ordinary number fractions exactly; the key extra skill is factorising so you can spot and cancel shared factors.

How do you simplify an algebraic fraction?

Simplifying means cancelling common factors from the numerator and denominator. Factorise both first, then cancel.

Worked example 1 — simplify 6x²y / (9xy²):

Numerator: 6x²y. Denominator: 9xy². Cancel common factors:

  • Numbers: 6 and 9 share factor 3 → 6/9 = 2/3.
  • x: x²/x = x.
  • y: y/y² = 1/y.

Answer: 2x / 3y

Worked example 2 — simplify (x² − 4) / (x + 2):

  1. Factorise the numerator: x² − 4 = (x + 2)(x − 2). (Difference of two squares.)
  2. The denominator is (x + 2).
  3. Cancel the (x + 2) factor: (x + 2)(x − 2) / (x + 2) = (x − 2).

Important: You can only cancel a factor — an expression that is multiplied. Never cancel terms that are added or subtracted. For example, (x + 4) / 4 cannot be simplified to x/1 + 1 = x + 1; the 4 in the denominator does not divide x and 4 separately.

How do you multiply algebraic fractions?

Multiply numerators together and denominators together, then simplify. Cancel before multiplying where possible.

Worked example — multiply (3x / 5) × (10 / x²):

  1. Multiply numerators: 3x × 10 = 30x.
  2. Multiply denominators: 5 × x² = 5x².
  3. Simplify: 30x / 5x² = 6 / x.

Alternatively, cancel first: 10/5 = 2 and x/x² = 1/x → (3 × 2) / x = 6/x. Same result, less arithmetic.

How do you divide algebraic fractions?

Dividing by a fraction is the same as multiplying by its reciprocal. Flip the second fraction and then multiply.

Worked example — divide (4x² / 3) ÷ (2x / 9):

  1. Flip the second fraction: 9 / (2x).
  2. Multiply: (4x² / 3) × (9 / 2x) = (4x² × 9) / (3 × 2x) = 36x² / 6x = 6x.

How do you add and subtract algebraic fractions?

The method is identical to adding ordinary fractions: find a common denominator, convert each fraction, then add or subtract the numerators.

Worked example 1 — add 1/x + 1/3:

  1. Common denominator = 3x.
  2. 1/x = 3/(3x) and 1/3 = x/(3x).
  3. Sum = (3 + x) / (3x) = (x + 3) / 3x.

Worked example 2 — subtract 5/(x + 2) − 3/(x − 1):

  1. Common denominator = (x + 2)(x − 1).
  2. Convert: 5(x − 1) / [(x + 2)(x − 1)] − 3(x + 2) / [(x + 2)(x − 1)].
  3. Numerator: 5(x − 1) − 3(x + 2) = 5x − 5 − 3x − 6 = 2x − 11.
  4. Answer: (2x − 11) / [(x + 2)(x − 1)].
Operation Rule Key step
Simplify Cancel common factors Factorise first
Multiply Multiply top × top, bottom × bottom Cancel before multiplying
Divide Flip second fraction, then multiply "KFC" — Keep, Flip, Change
Add/Subtract Find common denominator Expand numerators carefully

What mistakes do students commonly make?

  • Cancelling terms instead of factors. In (x + 3) / (x + 5), you cannot cancel the x — x is a term (added), not a factor (multiplied). Factorising first tells you what you can legally cancel.
  • Wrong common denominator. For 2/(x+1) + 3/(x+2), use the denominator (x+1)(x+2) — not just "x" or "1".
  • Sign errors when subtracting. When subtracting a fraction, every term in the numerator of the subtracted fraction changes sign. In example 2 above, −3(x + 2) = −3x − 6, not −3x + 6.

Frequently asked questions

Why do I need to factorise before simplifying?

Because you can only cancel a factor — something that multiplies the entire expression. Factorising reveals whether the numerator and denominator share a common bracket. Without factorising, you cannot see what can legally be cancelled.

Can an algebraic fraction ever equal zero?

The value of the fraction is zero when the numerator equals zero and the denominator is not zero. For example, (x − 3)/(x + 1) = 0 when x = 3. However, the fraction is undefined (not zero) when the denominator is zero — here when x = −1.

When does this topic appear on the GCSE paper?

Algebraic fractions appear on Higher tier papers, often in multi-step algebra questions. They are commonly combined with equation-solving: for example, "solve 3/x + 1/(x+1) = 2" requires finding a common denominator, multiplying through, and then solving the resulting linear or quadratic equation.

Both topics involve spotting and cancelling shared factors. With surds, you look for square number factors; with algebraic fractions, you factorise expressions. The underlying logic — cancel what is common, leave what is not — is the same.

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