Roots / Zeros
The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero, corresponding to x-intercepts on the graph.
Formula
f(x) = 0
Definition
The roots, or zeros, of a polynomial $f(x)$ are the x-values that make it equal zero, solutions to $f(x) = 0$; on a graph these are the x-intercepts, and "zero" and "root" describe exactly the same idea, plugging a root in gives $f(\text{root}) = 0$. Roots can be rational, irrational, or complex, and the factored form $f(x) = a(x - r_1)(x - r_2)$ shows them directly, connecting equations, functions, and factoring into a single unified view. Formally, a root $r$ of $f$ in $F[x]$ satisfies $f(r) = 0$ in $F$ or an extension field, and by the Factor Theorem, $r$ is a root exactly when $(x - r)$ divides $f(x)$; the multiplicity of a root is the largest $k$ such that $(x - r)^k$ divides $f(x)$, and the Fundamental Theorem of Algebra guarantees exactly $n$ roots (with multiplicity) for a degree-$n$ polynomial over $\mathbb{C}$.
Example
$f(x) = x^2 - 9$: solving $x^2 - 9 = 0$ gives $x = 3$ or $x = -3$, where the graph crosses the x-axis. $f(x) = 2x^2 - 8 = 2(x+2)(x-2)$ has roots $x = 2$ and $x = -2$. For $f(x) = (x-2)^2(x+1)$, the root $x = 2$ has multiplicity $2$ (a double root) and $x = -1$ has multiplicity $1$; the multiplicities sum to $3$, matching the degree of $f$.
Key Insight
Roots link three views of the same polynomial: as an equation to solve, as a function's x-intercepts, and as the linear factors in a factorization. Root multiplicity shapes local graph behavior, a simple root crosses the x-axis, an even-multiplicity root touches and bounces off, an odd-multiplicity root crosses while flattening out, a distinction visible on any graph and fundamental to algebraic geometry.