Let $p$ be the quartic (degree 4) polynomial that satisfies $$p(i) = 2^i,\quad i = 0,1,2,3,4.$$ Then
$p(x) =$ .
Hint: You may have a better idea, but a brute force approach is to write $$p(x) = ax^4 + bx^3 + cx^2 + dx + e$$ where $a$, $b$, $c$, $d$, and $e$, are the unknown coefficients, and then solve the linear system $p(0) = 1$, $p(1) = 2$, $p(2) = 4$, $p(3) = 8$, and $p(4) = 16$ for $a$, $b$, $c$, $d$, and $e$.