In Zeros and Factored Form each factor gave a zero where the graph crossed the axis. When a factor is repeated, something new happens. You also know from End Behavior of Polynomials what happens far away; now look closely at each zero.
Idea clave
If $(x - r)^m$ is a factor (and no higher power is), $r$ is a zero of multiplicity $m$. Odd multiplicity: the graph crosses the $x$-axis at $r$. Even multiplicity: it touches the axis and turns back. A multiplicity above 1 also flattens the graph near the zero.
Por qué funciona
Near $x = r$, the factor $(x - r)^m$ changes much faster than the other factors, which stay close to a fixed nonzero number. So near $r$ the graph looks like $c(x - r)^m$. If $m$ is even, $(x - r)^m$ is never negative, so the graph stays on one side: it touches. If $m$ is odd, $(x - r)^m$ changes sign, so the graph crosses. A higher power is very small near 0 (like $0.1^3 = 0.001$), which is why the graph flattens.
Worked examples
Ejemplo resuelto: Reading multiplicities
Describe the zeros of $f(x) = (x - 3)^2(x + 1)^3 x$.
- $x = 3$ has multiplicity 2 (even): the graph touches and turns back.
- $x = -1$ has multiplicity 3 (odd, above 1): it crosses but flattens, like $x^3$ at 0.
- $x = 0$ has multiplicity 1: it crosses like a line.
Respuesta: Touch at 3; flat crossing at $-1$; plain crossing at 0.
Ejemplo resuelto: Counting
What is the degree of $f$ above, and how many distinct zeros does it have?
- Add the multiplicities: $2 + 3 + 1 = 6$.
- Distinct zeros: $3$, $-1$ and $0$, so three.
Respuesta: Degree 6, three distinct zeros.
Error común
Thinking every zero is a crossing point. The graph of $(x - 3)^2$ only touches the axis at 3. Check with numbers on each side: $(2.9 - 3)^2$ and $(3.1 - 3)^2$ are both positive, so the graph doesn't change sides at 3.