Definition and degree of a polynomial

What is a polynomial?

A polynomial has one variable, x, and real (or complex) coefficients, and is written in the form: P(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0 where the a_i are fixed scalars, and a_n is called the leading coefficient when a_n ≠ 0.

Degree and leading coefficient

The degree deg(P) is the largest exponent n such that a_n ≠ 0. For example, P(x) = 3x⁴ - 2x² + x - 5 has degree 4 and leading coefficient 3.

Polynomial	Degree	Leading coefficient
5	0	5
2x - 7	1	2
x³ + x	3	1
0 (zero polynomial)	not defined	-

Common pitfall

The zero polynomial has no degree in the usual sense. By convention, it is assigned -∞ so that the rule deg(P*Q) = deg(P) + deg(Q) remains valid even when one of the factors is zero.

Operations and degree rules

Addition is carried out coefficient by coefficient: we always have deg(P+Q) ≤ max(deg P, deg Q), with equality holding unless terms of higher degree cancel out. Multiplication utilises distributivity; for example, (x+1)(x²-x+1) = x³+1. Over R or C (integral rings), we have exactly deg(P*Q) = deg(P) + deg(Q), a very useful property for verifying a factorisation.