Term 1 review: binomial expansions, complex numbers, and vectors, lines and planes

Weeks 1 to 3 give one theorem and several ways of using it. (x + a)ⁿ has n + 1 terms and its general term is T(r+1) = nCr xⁿ⁻ʳ aʳ, with r counted from 0. Read the question first: ‘term’ includes its power of x, ‘coefficient’ is the number in front, and ‘power’ is only the exponent. For a term counted from the end, convert it to a place counted from the start. For terms with a given power of x, write T(r+1) with every part of a in brackets, simplify the power of x, set it equal to k, solve for r, and accept r only if it is a whole number from 0 to n; the free term is the case k = 0. The middle term depends on whether n is even (one) or odd (two). Ratios use T(r+1)/T(r) = ((n − r + 1)/r) · (a/x); to find the greatest term, solve that ratio ≥ 1 for r, and when signs alternate compare the sizes first. Sums of coefficients come from x = 1. Keep the table as a checklist: before computing, decide which row your question belongs to.