The binomial theorem

Learning objectives

Expand (a + b)ⁿ for a positive integer n using the binomial theorem, obtaining the coefficients from Pascal's triangle or from ⁿCr

Write down the general term ⁿCr a^(n − r) bʳ of an expansion and use it to find a specific term or the coefficient of a stated power of x without expanding fully

Find the term independent of x, by setting the total power of x in the general term equal to zero and solving for r

Expand brackets such as (2 + 3x)ⁿ or (x − 2/x)ⁿ, dealing carefully with the powers of the numerical coefficients and with negative signs

Find an unknown constant in an expansion from information about a coefficient, and find a coefficient in a product such as (1 + x)(2 − x)ⁿ by combining the relevant terms