Degree of a Polynomial
AlgebraThe degree of a polynomial is the value of the largest exponent on the variable in the polynomial, which determines its general shape and behavior.
Definition
The degree of a polynomial is the biggest exponent in the polynomial. It tells you the "power level" of the expression.
Example
$5x^3 + 2x - 7$ has degree $3$ (highest exponent is $3$). $4x^2 + x$ has degree $2$. The number $8$ alone is degree $0$ (think of it as $8x^0$).
Key Insight
The degree tells you how the polynomial curves. Degree $1$ is a straight line, degree $2$ is a U-shape (parabola), and higher degrees have more curves.
Definition
The degree of a polynomial in one variable is the largest exponent of the variable with a non-zero coefficient. For a polynomial in multiple variables, the degree of each term is the sum of all variable exponents in that term, and the polynomial's degree is the largest such sum.
Example
$7x^4 - 3x^2 + x - 1$ has degree $4$. $3x^2y + 2xy^3 - y$: the degrees of terms are $3, 4, 1$; the polynomial has degree $4$.
Key Insight
The degree controls the end behavior of the polynomial: even-degree polynomials go to the same end in both directions (both up or both down); odd-degree polynomials go to opposite ends.
Definition
The degree function $\deg: F[x] \to \mathbb{Z}_{\ge 0} \cup \{-\infty\}$ (with $\deg(0) = -\infty$) satisfies $\deg(fg) = \deg(f) + \deg(g)$ and $\deg(f+g) \le \max(\deg(f), \deg(g))$. This makes $F[x]$ a graded ring. The degree bounds the number of roots: a non-zero polynomial of degree $n$ has at most $n$ roots in any field $F$ (more precisely, exactly $n$ roots in the algebraically closed field).
Example
A degree-$5$ polynomial over $\mathbb{R}$ has at most $5$ real roots, and exactly $5$ complex roots (counted with multiplicity). The polynomial $x^5 - 1$ has $1$ real root and $4$ complex roots.
Key Insight
The degree of a polynomial directly bounds the number of intersections between its graph and any horizontal line. This makes degree a fundamental invariant in intersection theory and algebraic geometry.