> ## Content Index > Fetch the complete content index at: https://www.esheets.io/llms.txt > Use this file to discover other available public pages before exploring further. # Simplifying Algebraic Fractions - URL: https://www.esheets.io/simplifying-algebraic-fractions/ - Published: 2026-10-06T18:57:14.000Z - Updated: 2026-10-06T18:57:14.000Z - Author: Richard Linnington - Tags: Maths, Algebra 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](#practise-now) ## Practise now 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](#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.