Simplifying Algebraic Fractions
Practise simplifying algebraic fractions by factorising the numerator and denominator, then cancelling common factors. This self-marking GCSE Higher worksheet builds towards Grade 7, from coefficient and index-law cancellation to quadratic fractions with sign changes and non-monic quadratics.
Use your knowledge of expanding brackets and factorising quadratics to give each answer in its simplest form. Jump to the questions
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Worksheet preview and key skills
Worksheet preview
Eight questions progress from cancelling coefficients and powers to factorising harder quadratic fractions. Each question is worth one mark.
Enter your simplified numerator and denominator separately, using denominator 1 for an expression. Equivalent answers that still have a common factor need further simplification.
What you’ll practise
- Reducing coefficients and powers, and taking out common monomial factors.
- Cancelling matching brackets and factorising linear expressions.
- Factorising monic quadratics and differences of two squares, with careful signs.
- Factorising non-monic quadratics in Grade 7 algebraic fractions.
Use the interactive worksheet below, or read the Topic guide for the method and worked example.
Simplifying Algebraic Fractions
Simplify each fraction fully. Enter the numerator and denominator separately; use denominator 1 for a non-fraction answer. Use the ², ³ or ⁴ buttons for powers; add brackets where needed.
Topic guide
An algebraic fraction is simplified by cancelling common factors of the whole numerator and denominator. First simplify any terms, then factorise, identify matching factors and cancel. Finally check that no numerical or algebraic common factor remains.
1. Coefficients and powers
Reduce the coefficient fraction and subtract powers of the same letter when dividing.
14x435x2 = 2x25. Here 14 ÷ 35 reduces to 2 ÷ 5, and x4 ÷ x2 = x2.
2. Take out a common factor
Factorise the whole numerator before cancelling. Every term must be included.
15x2 + 20x5x = 5x(3x + 4)5x = 3x + 4.
3. Matching brackets and linear expressions
A complete bracket can be a common factor.
(x + 6)(x − 3)(x + 6)(x + 9) = x − 3x + 9. Cancel the factor (x + 6).
Sometimes the matching brackets appear only after factorising both expressions:
8x + 4014x + 70 = 8(x + 5)14(x + 5) = 47.
The original denominator must be non-zero, and its excluded values remain excluded after simplification; in the example above, x ≠ −5 still applies even though the simplified fraction is 4/7.
4. Monic quadratics
To factorise x2 + bx + c, find two integers whose sum is b and whose product is c.
5x + 10x2 + 9x + 14 = 5(x + 2)(x + 2)(x + 7) = 5x + 7.
When both sides are quadratic, factorise each one separately:
x2 − 3x − 18x2 − x − 30 = (x − 6)(x + 3)(x − 6)(x + 5) = x + 3x + 5.
5. Difference of two squares and signs
Use a2 − b2 = (a − b)(a + b). Reversing a subtraction changes its sign: 4 − x = −(x − 4).
16 − x2x2 + 3x − 28 = −(x − 4)(x + 4)(x − 4)(x + 7) = −x − 4x + 7. The minus sign remains after cancellation.
6. Harder quadratic factorisation
For ax2 + bx + c with a > 1, look for two numbers with product ac and sum b. Split the middle term and factorise by grouping. For example, 2x2 + 5x − 3 = 2x2 + 6x − x − 3 = (2x − 1)(x + 3).
2x2 + 5x − 36x2 + x − 2 = (2x − 1)(x + 3)(2x − 1)(3x + 2) = x + 33x + 2.
Common mistakes
- Cancel factors, not separate terms. In x + 3x, x is not a factor of the whole numerator, so it cannot be cancelled.
- Do not cancel just part of a bracket or lose a minus sign when reversing a subtraction.
- Reduce numerical coefficients as well as algebraic factors. Check the index laws when cancelling powers.
- Do not stop at an equivalent fraction if a common factor still remains.
This worksheet simplifies one algebraic fraction at a time. Adding, subtracting, multiplying and dividing separate algebraic fractions are different skills.