Answer
$x^3-2x^2-8x$
$x^4-2x^3-8x^2$
$x^5-2x^4-8x^3$
Work Step by Step
A cubic function with $3$ given $x$-intercepts $x_1,x_2,x_3$ can be written in the form:
$$f(x)=a(x-x_1)(x-x_2)(x-x_3),$$
where $a$ is an arbitrary real number.
For example, for $a=1$:
$$\begin{align*}
f(x)&=(x-(-2))(x-0)(x-4)\\
&=(x+2)x(x-4)\\
&=(x^2+2x)(x-4)\\
&=x^3-4x^2+2x^2-8x\\
&=x^3-2x^2-8x.
\end{align*}$$
A quartic function with $3$ given $x$-intercepts $x_1,x_2,x_3$ can be written in the form:
$$f(x)=a(x-x_1)^2(x-x_2)(x-x_3),$$
where $a$ is an arbitrary real number.
For example, for $a=1, x_1=0, x_2=-2,x_3=4$:
$$\begin{align*}
f(x)&=(x-0)^2(x-(-2))(x-4)\\
&=x^2(x+2)(x-4)\\
&=x^2(x^2-2x-8)\\
&=x^4-2x^3-8x^2.
\end{align*}$$
A fifth-degree function with $3$ given $x$-intercepts $x_1,x_2,x_3$ can be written in the form:
$$\begin{align*}
f(x)&=a(x-x_1)^2(x-x_2)^2(x-x_3)\text{ or}\\
f(x)&=a(x-x_1)^3(x-x_2)^2(x-x_3).
\end{align*}$$
where $a$ is an arbitrary real number.
For example, for $a=1, x_1=0, x_2=-2,x_3=4$:
$$\begin{align*}
f(x)&=(x-0)^3(x-(-2))(x-4)\\
&=x^3(x+2)(x-4)\\
&=x^3(x^2-2x-8)\\
&=x^5-2x^4-8x^3.
\end{align*}$$
