Polynomial zeros and multiplicity

Parametric equations

A polynomial in factored form:

f (x) = a (x - r_{1})^{m_{1}} (x - r_{2})^{m_{2}} (x - r_{3})^{m_{3}}

Each $r_{i}$ is a zero, and each multiplicity $m_{i}$ is $1$ or $2$ . At $x = r_{i}$ , $f (r_{i}) = 0$ , so the graph meets the x-axis there.

The equation $f (x) = 0$ asks where the zeros are. The inequality $f (x) > 0$ asks which intervals sit above the x-axis.

Three crossings / One double root / Two touches: jump among three simple zeros, one double root in the middle, and two touches
Leading coefficient a: change the leading coefficient. The sign of $a$ flips which intervals are positive
Zero r₁, Zero r₂, and Zero r₃: drag a zero, or drag its mark on the figure
Multiplicity m₁, Multiplicity m₂, and Multiplicity m₃: each switch is only $1$ or $2$ . Multiplicity $1$ crosses the x-axis. Multiplicity $2$ touches and turns back
Multiplicity guide: mark each zero as crosses or touches. The sign line under the graph stays on: gold where $f (x) > 0$ , gray where $f (x) < 0$

The zeros cut the number line into intervals. The sign on an interval comes from the leading coefficient $a$ together with the multiplicities.
Multiplicity $1$ crosses the x-axis. Multiplicity $2$ only touches, so the sign is the same on both sides.
A quadratic is the special case of at most two zeros, each of multiplicity $1$ or $2$ . The discriminant counts the real roots. The multiplicity says whether the graph crosses or touches.