Algebra - Binomial theorem — Binomial theorem for integer positive power

About this lesson

This unit teaches the binomial theorem for positive integer powers. It begins by relating Pascal's triangle to the coefficients of binomial expansions, showing how the coefficients can be expressed using combinations (e.g., nCr). The unit then presents the general expansion formulas for (x + a)^n and (x - a)^n, along with remarks on the number of terms, ordering, and sum of powers. It covers special cases like (1 + x)^n and (1 - x)^n, and introduces the general term T_{r+1} = nCr x^{n-r} a^r. Students learn to find specific terms, including terms from the end, and to use rules for combining expansions like (x + a)^n + (x - a)^n. The unit also explains how to determine the middle term(s) in expansions, depending on whether n is even or odd. Worked examples and exercises involve finding coefficients, solving for unknowns, and proving identities. By the end, students should be able to expand binomials, find any term, identify middle terms, and apply the theorem to solve problems.