Roots / Zeros
AlgebraThe 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 are the x-values that make the polynomial equal to zero. On a graph, they are the x-intercepts, where the curve crosses or touches the x-axis.
Example
$f(x) = x^2 - 9$: roots are found by solving $x^2 - 9 = 0$, so $x^2 = 9$, $x = 3$ or $x = -3$. The graph crosses the x-axis at $x = 3$ and $x = -3$.
Key Insight
"Zero" and "root" mean the same thing: the x-value that makes the function output zero. Plugging the root in gives $f(\text{root}) = 0$.
Definition
The roots (zeros) of a polynomial $f(x)$ are solutions to $f(x) = 0$. For a quadratic, there are at most two roots. Roots can be rational, irrational, or complex (non-real). Real roots correspond to x-intercepts. The factored form $f(x) = a(x - r_1)(x - r_2)$ directly shows the roots $r_1$ and $r_2$.
Example
$f(x) = 2x^2 - 8 = 2(x^2 - 4) = 2(x+2)(x-2)$. Roots: $x = 2$ and $x = -2$. These are also the x-intercepts of the parabola $y = 2x^2 - 8$.
Key Insight
Roots connect equations (solutions) and functions (x-intercepts) and factoring (linear factors). The same concept unifies all three views of a polynomial.
Definition
A root (zero) of polynomial $f$ in $F[x]$ is an element $r$ in $F$ (or in a field extension of $F$) such that $f(r) = 0$. By the Factor Theorem, $r$ is a root iff $(x - r)$ divides $f(x)$. The multiplicity of a root $r$ is the largest $k$ such that $(x - r)^k$ divides $f(x)$. The Fundamental Theorem of Algebra guarantees that every non-constant polynomial over $\mathbb{C}$ has at least one root, hence exactly $n$ roots (counted with multiplicity) for a degree-$n$ polynomial.
Example
$f(x) = (x-2)^2(x+1)$. Root $x = 2$ has multiplicity $2$ (double root), root $x = -1$ has multiplicity $1$. Sum of multiplicities $= 3 =$ degree of $f$.
Key Insight
Root multiplicity determines local behavior: a simple root (multiplicity $1$) crosses the x-axis; an even-multiplicity root touches and bounces off; an odd-multiplicity root crosses with a flattening. This is visible on the graph and fundamental to algebraic geometry.