AMC 12 2017b Problem 23

The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is $24$. What is $f(0)$?

\textbf{(A)}\ -2 \textbf{(B)}\ 0 \textbf{(C)}\ 2 \textbf{(D)}\ \frac{24}{5} \textbf{(E)}\ 8