Maths: Binomial Expansions Build Polynomial Products

The binomial theorem gives a fast route to expanding expressions like (3 − 2x)⁵ without multiplying everything out by hand. Each term follows the pattern nCr · a^(n−r) · b^r, where the binomial coefficients nCr count the ways of choosing r factors of the second term, and the powers of a and b shift in opposite directions as you move along the expansion. This structure turns a seemingly long product into a tidy sum of ascending powers of x. That same structure is what makes binomial expansions so useful elsewhere. Once the first few terms are known, they can be fed straight into a second product, such as (1 + x)(3 − 2x)⁵, where the coefficient of any power of x comes from adding together every pair of terms whose powers combine to give that power. Spotting which products contribute, and how the coefficients and powers interact, is the real skill underlying both polynomial multiplication and the binomial theorem itself.