Answer
$-480$
Work Step by Step
A cubic polynomial function can be written as:
$$f(x)=ax^3+bx^2+cx+d.$$
Since the leading coefficient is $2$, it means $a=2$. Since the constant term is $-5$, it means $d=-5$.
$$f(x)=2x^3+bx^2+cx-5.$$
Now we construct a system of equations using $ f(1)=0$ and $f(2)=3$:
$$\begin{cases}
2(1^3)+b(1^2)+c(1)-5=0\\
2(2^3)+b(2^2)+c(2)-5=3.
\end{cases}$$
$$\begin{cases}
2+b+c-5=0\\
16+4b+2c-5=3.
\end{cases}$$
$$\begin{cases}
b+c=3\\
4b+2c=-8.
\end{cases}$$
Solve the system:
$$\begin{cases}
-2b-2c=-6\\
4b+2c=-8.
\end{cases}$$
$$2b=-14\Rightarrow b=-7$$
$$-7+c=3\Rightarrow c=10.$$
The polynomial is:
$$f(x)=2x^3-7x^2+10x-5.$$
Calculate $f(-5)$:
$$f(-5)=2(-5)^3-7(-5)^2+10(-5)-5=-480.$$
