The personalized polynomial

Text

A cubic “personalized polynomial” $p$ satisfies $p(0)=3$, $p(1)=1$, $p(2)=13$, $p(3)=5$. Using finite differences (for a third-degree polynomial the third difference is constant), compute $p(4)$.

Solution

Build the difference table from the values $3,\,1,\,13,\,5$:

• first differences: $-2,\ 12,\ -8$
• second differences: $14,\ -20$
• third difference: $-34$ (constant, since $p$ has degree $3$).

Extend keeping the third difference constant: $\Delta^2_{\text{next}}=-20+(-34)=-54,\quad \Delta^1_{\text{next}}=-8+(-54)=-62,$ $p(4)=5+(-62)=\boxed{-57}.$