Degree of a Polynomial

Intuition and core definition

For a nonzero polynomial in one variable, the degree is the greatest exponent with a nonzero coefficient after simplification. A nonzero constant has degree $0$. The zero polynomial has no universally assigned finite degree; it is often described as having undefined degree so standard degree rules remain consistent.

Notation, language, and conditions

In $P(x)=a_nx^n+\cdots+a_0$ with $a_n\ne0$, $n=\deg P$. For a multivariable term, total degree is the sum of exponents; polynomial degree is the greatest total degree among terms. Some contexts also discuss degree in one chosen variable, so the convention should be named.

Why this idea matters

Polynomial degree records the largest exponent with a nonzero coefficient and predicts broad features such as end behaviour and the maximum possible number of roots.

A dependable method

1. Expand or combine like terms enough to reveal cancellations.
2. Remove zero-coefficient terms.
3. For one variable, inspect the largest remaining exponent.
4. For several variables, add exponents within each term and take the greatest total.
5. State special handling for the zero polynomial.

Worked example

Common mistakes