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This topic extends the binomial expansion of (1 + x)ⁿ to negative and fractional powers, giving an infinite series valid for |x| < 1, used for approximations and with partial fractions. 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
For negative or fractional powers the expansion is infinite. Write the bracket as 1 plus something first, factorising out any constant, then apply the formula. The expansion is only valid when the size of that something is less than 1, and stating that range is worth a mark.
Before you start
Before starting, revisit the binomial expansion for positive indices. It sets up expansions of rational functions and partial fractions.
What you learn
Sometimes a function is a linear (or polynomial) factor times a binomial. The course works through the binomial for rational n, expansions of (1+bx)ⁿ and validity, expansions of (a+bx)ⁿ, approximations and percentage error, partial fractions: distinct, improper, repeated and A non-binomial factor that just multiplies. 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
Binomial Expansion (C34) is a Sequences & Series topic in the A-level syllabus, covered here at the second year of A-level. It runs to 6 concepts, each followed by a check, with 6 worked examples along the way. It sits alongside Geometric Sequences, Geometric Series and Recurrence Relations in the same strand. Endless extra practice on this strand is available on the Sequences & Series revision engine, which generates a fresh question each time.
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