Polynomial Identities and the Binomial Theorem

A polynomial identity holds for every value. (a + b)² = a² + 2ab + b² is one, and the middle term is what people forget.

Sums and differences of cubes

a³ − b³ = (a − b)(a² + ab + b²) and a³ + b³ = (a + b)(a² − ab + b²). The sign inside the bracket opposes the outer one.

Pascal's triangle

Each row begins and ends with 1, and every other entry is the sum of the two above it. Row n holds the coefficients of (a + b)ⁿ.

They are combinations

The entries of row n are nC0 through nCn, because choosing which factors contribute a b is literally a combination.

The exponent pattern

The power of a falls from n to 0 while the power of b rises. Every term totals n.

One term at a time

The term with bʳ is nCr · a^(n−r) · bʳ, which finds a single term without expanding the rest.

An identity holds for every value

A polynomial identity is true for all inputs, unlike an equation which is true for some. Proving one means transforming one side into the other, not finding values that work.

Expanding a power directly

(x + y)ⁿ expands into n + 1 terms with binomial coefficients. Rather than multiplying n brackets, the theorem writes the answer, which matters as soon as n exceeds three or four.

The coefficients are combinations

The coefficient of xᵏyⁿ⁻ᵏ is n choose k, because expanding means choosing x from k of the n brackets. The counting and the algebra are the same computation.

Extracting one term

The general term formula gives, say, the x⁵ coefficient of a twelfth power without expanding anything else. That selective extraction is where the theorem earns its keep.