Roots of a polynomial and the factorisation theorem

Root of a polynomial

A number a (real or complex) is a root of P if P(a) = 0. The factorisation theorem states that, in this case, P(x) is divisible by (x - a): P(x) = (x - a) * Q(x), where deg(Q) = deg(P) - 1

Multiplicity of a root

a is a root of multiplicity k if P(x) = (x - a)^k * Q(x), where Q(a) ≠ 0. We speak of a simple root if k = 1, a double root if k = 2, and so on. We can test the multiplicity using successive derivatives: a is a root of multiplicity ≥ k if and only if P(a) = P'(a) = … = P^(k-1)(a) = 0.

Example

P(x) = x³ - 3x² + 3x - 1 = (x-1)³. Here, a = 1 is a triple root: P(1) = 0, P'(1) = 0, P''(1) = 0, but P'''(1) = 6 ≠ 0, which confirms a multiplicity exactly equal to 3.

Number of roots

A polynomial of degree n has at most n distinct real roots. Over the complex numbers, it has exactly n roots, counting multiplicities (Fundamental Theorem of Algebra).

Common pitfall

After partial factoring, never assume that a value which cancels out an intermediate factor is necessarily a root of the full polynomial: you must always substitute this value back into the original P(x) to check that P(a) = 0.