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The factor theorem states (x − a) is a factor of f(x) exactly when f(a) = 0, and the remainder theorem gives the remainder as f(a). Together they factorise cubics and solve polynomial equations. It's a key topic in A-Level and International A-Level (iAL) Pure Mathematics. This free interactive eClassroom course walks through each step with worked examples and instant-feedback questions, so you can practise until it's secure.
How you do it
If substituting a value gives zero, that value is a root and the corresponding bracket is a factor. Otherwise the value you get is the remainder. Test the factors of the constant term to find a root, then divide out to get a quadratic.
Before you start
This builds directly on polynomial substitution. It sets up algebraic division and factorising cubics. Pure 2 is where it is assessed.
What you learn
Two powerful shortcuts based on evaluating a polynomial. Together they let you find remainders, test factors, pin down unknown coefficients, and factorise and solve cubics — no long division needed. Each factor or remainder condition gives one equation. The course works through the remainder theorem, the factor theorem, finding unknown coefficients, fully factorising a cubic and solving cubics & ‘show that’. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
We teach Factor & Remainder Theorem as one of the Algebra topics in our A-level course library, at the AS year. It runs to 5 concepts, each followed by a check, with 5 worked examples along the way. Nearby topics in the same strand are Functions - Domain and Range, Improper Algebraic Fractions and Indices and Surds. For unlimited practice, the Algebra revision engine builds new questions on demand and records your accuracy.
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