Question

given the graph of a degree 4 polynomial below, complete the table of values for either the x - value of zero, or the multiplicity of the zero.
root with x = | multiplicity
—4 |
| 1
|
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Explanation:

Step1: Identify zeros from x-intercepts

The graph crosses/touches the x-axis at $x=-4$, $x=-1$, and $x=4$.

Step2: Determine multiplicity of $x=-4$

The graph crosses the x-axis at $x=-4$, so multiplicity is odd. Since it does not flatten here, multiplicity = 1.

Step3: Identify root with multiplicity 1

The graph crosses the x-axis at $x=4$ (no flattening), so this is the root with multiplicity 1.

Step4: Analyze remaining root $x=-1$

The graph touches the x-axis at $x=-1$ (bounces), so multiplicity is even. The polynomial is degree 4: $1+1+m=4$, so $m=2$.

Answer:

Root with x =	Multiplicity
$4$	$1$
$-1$	$2$