Degree of a Polynomial
The 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 its biggest exponent: the largest exponent of the variable with a non-zero coefficient (a constant alone, like $8$, has degree $0$, thought of as $8x^0$). 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. The degree controls end behavior: degree $1$ graphs a straight line, degree $2$ a U-shaped parabola, even degrees send both ends of the graph in the same direction, and odd degrees send the ends in opposite directions. Formally, 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 $\deg(f+g) \le \max(\deg(f), \deg(g))$, making $F[x]$ a graded ring, and it bounds the number of roots: a non-zero degree-$n$ polynomial has at most $n$ roots in any field, exactly $n$ in an algebraically closed one.
Example
$5x^3 + 2x - 7$ has degree $3$, and $4x^2 + x$ has degree $2$. $7x^4 - 3x^2 + x - 1$ has degree $4$; for $3x^2y + 2xy^3 - y$, the term degrees are $3, 4, 1$, so the polynomial has degree $4$. A degree-$5$ polynomial over $\mathbb{R}$ has at most $5$ real roots and exactly $5$ complex roots (counted with multiplicity); $x^5 - 1$ has $1$ real root and $4$ complex roots.
Key Insight
The degree tells you the "power level" of the expression and how it curves. This same idea, the degree bounding the number of intersections between a graph and any horizontal line, makes degree a fundamental invariant throughout intersection theory and algebraic geometry.