Algebraic fractions and division

Simplifying

Factorise the top and the bottom fully, then cancel common factors. You can never cancel single terms: (x + 6)/(x + 3) does not simplify to 2.

(x² - 9)/(x² + x - 6) = (x - 3)(x + 3)/((x + 3)(x - 2)) = (x - 3)/(x - 2).

Multiplying, dividing, adding

• Multiply: factorise, cancel, then multiply tops and bottoms.
• Divide: multiply by the reciprocal of the second fraction.
• Add or subtract: use a common denominator, ideally the lowest one, then simplify the numerator.

Algebraic division

If the degree of the top is at least the degree of the bottom, the fraction is improper. Divide to write it as quotient + remainder/divisor:

(x³ + 2x² - 5x + 4)/(x - 1) = x² + 3x - 2 + 2/(x - 1).

When dividing by a quadratic, the remainder can be linear, of the form (Cx + D).

Worked example

Simplify (x² - 9)/(x² + x - 6).

1. Factorise the top: (x - 3)(x + 3).
2. Factorise the bottom: (x + 3)(x - 2).
3. Cancel the common factor (x + 3).

Answer: (x - 3)/(x - 2)

Check yourself

1. Simplify (2x² + 5x - 3)/(4x² - 1).
   • (x + 3)/(2x + 1)
   • (x + 3)/(2x - 1)
   • (2x + 3)/(4x + 1)
   • (x - 3)/(2x + 1)
   Show answer

   (x + 3)/(2x + 1). Top: (2x - 1)(x + 3). Bottom: (2x - 1)(2x + 1). Cancel (2x - 1).

2. Simplify (x + 2)/(x - 1) ÷ (x² - 4)/(x² - 1).
   • (x - 1)/(x + 2)
   • (x + 2)²/(x - 1)²
   • (x + 1)/(x - 2)
   • (x + 1)/(x + 2)
   Show answer

   (x + 1)/(x - 2). Multiply by the reciprocal: (x + 2)/(x - 1) × (x - 1)(x + 1)/((x - 2)(x + 2)). Cancel (x + 2) and (x - 1).

3. Write 3/(x + 1) + 2/(x - 2) as a single fraction.
   • (5x - 4)/((x + 1)(x - 2))
   • (5x + 4)/((x + 1)(x - 2))
   • 5/((x + 1)(x - 2))
   • 5/(2x - 1)
   Show answer

   (5x - 4)/((x + 1)(x - 2)). Common denominator (x + 1)(x - 2). Numerator: 3(x - 2) + 2(x + 1) = 3x - 6 + 2x + 2 = 5x - 4.

4. Write (x³ + 2x² - 5x + 4)/(x - 1) in the form Ax² + Bx + C + D/(x - 1). Find D.
   Show answer

   2. Dividing gives x² + 3x - 2 with remainder 2, so D = 2. (Check: f(1) = 1 + 2 - 5 + 4 = 2.)

5. Write (2x³ - x² + 4x - 7)/(x² + 1) in the form Ax + B + (Cx + D)/(x² + 1). Find the constant D.
   Show answer

   -6. 2x³ - x² + 4x - 7 = (2x - 1)(x² + 1) + (2x - 6). So A = 2, B = -1, C = 2 and D = -6.

6. A student writes (x² + 4)/(x + 2) = x + 2. What is wrong?
   • Nothing: x² + 4 = (x + 2)²
   • The answer should be x - 2
   • The answer should be x² + 2
   • x² + 4 does not factorise, so nothing cancels
   Show answer

   x² + 4 does not factorise, so nothing cancels. Only factors cancel. x² + 4 has no real factors, so (x + 2) is not a factor of the top.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.