Skill: Identify repeated zeros and their graphical behaviour

A zero is an $x$-value that makes the polynomial equal to $0$. In factored form, the exponent on a factor tells you the zero’s multiplicity.

• An odd multiplicity means the graph crosses the $x$-axis.
• An even multiplicity means the graph touches the $x$-axis and turns around.
• A larger multiplicity usually makes the graph look flatter near the zero.
• A repeated zero is one zero with a factor occurring more than once, not several different intercepts.

Example

Analyze the real zeros and describe the graph of

$$f(x)=-2(x+1)^3(x-2)^4(x-5).$$

Step 1: Set each factor equal to zero.

$$x+1=0 \quad\Rightarrow\quad x=-1$$ $$x-2=0 \quad\Rightarrow\quad x=2$$ $$x-5=0 \quad\Rightarrow\quad x=5$$

So the real zeros are $x=-1$, $x=2$, and $x=5$.

Step 2: Read the multiplicity from each exponent.

• $(x+1)^3$ gives $x=-1$ with multiplicity $3$.
• $(x-2)^4$ gives $x=2$ with multiplicity $4$.
• $(x-5)$ gives $x=5$ with multiplicity $1$.

The zero $x=-1$ is one intercept, even though its factor occurs three times. Similarly, $x=2$ is one intercept with multiplicity $4$.

Step 3: Use the parity of each multiplicity.

Zero	Multiplicity	Graph behavior
$x=-1$	$3$	Crosses the $x$-axis; the odd multiplicity can make it look somewhat flat
$x=2$	$4$	Touches the $x$-axis and turns; the even multiplicity makes it flatter
$x=5$	$1$	Crosses the $x$-axis normally

Therefore, the graph crosses at $x=-1$ and $x=5$, but touches and turns at $x=2$. The zero at $x=2$ has the larger multiplicity, so the graph is especially flat near that intercept.