To determine the polynomial [tex]\( h(x) \)[/tex] from the given table, we use the information provided: every [tex]\( x \)[/tex]-intercept of [tex]\( h(x) \)[/tex] is shown in the table and has a multiplicity of one.
Here are the steps to solve this:
1. Identify the [tex]\( x \)[/tex]-intercepts from the table:
We are given:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & -2 & 0 & 1 & 2 & 3 \\
\hline
h(x) & 0 & -12 & 0 & 8 & 0 \\
\hline
\end{array}
\][/tex]
The [tex]\( x \)[/tex]-intercepts are the values of [tex]\( x \)[/tex] where [tex]\( h(x) = 0 \)[/tex], which are:
[tex]\[
x = -2, 1, 3
\][/tex]
2. Form the polynomial based on its roots:
Since each [tex]\( x \)[/tex]-intercept is shown in the table and has a multiplicity of one, the polynomial can be constructed by multiplying the factors corresponding to each root:
[tex]\[
h(x) = k(x + 2)(x - 1)(x - 3)
\][/tex]
Given that the leading coefficient is 1, [tex]\( k \)[/tex] must be 1.
So, the polynomial takes the form:
[tex]\[
h(x) = (x + 2)(x - 1)(x - 3)
\][/tex]
3. Expand the polynomial:
[tex]\[
\begin{aligned}
h(x) &= (x + 2)(x - 1)(x - 3) \\
&= (x + 2)\left[(x - 1)(x - 3)\right] \\
&= (x + 2)\left[x^2 - 4x + 3\right] \\
&= x(x^2 - 4x + 3) + 2(x^2 - 4x + 3) \\
&= x^3 - 4x^2 + 3x + 2x^2 - 8x + 6 \\
&= x^3 - 2x^2 - 5x + 6
\end{aligned}
\][/tex]
4. Match the polynomial with the given options:
The expanded polynomial is:
[tex]\[
h(x) = x^3 - 2x^2 - 5x + 6
\][/tex]
Therefore, the correct polynomial that fits the given values and conditions is:
[tex]\[
h(x) = x^3 - 2x^2 - 5x + 6
\][/tex]
So, the equation of the polynomial function is:
[tex]\[
\boxed{x^3 - 2x^2 - 5x + 6}
\][/tex]
