Question

Suppose f is a polynomial such that f(0)=47, f(1)=32, f(2)=-13, and f(3)=16. What is the sum of the coefficients of f?

Answer 1

assuming that f(x) = ax^3+bx^2+cx+d,

d=47
a+b+c+47 = 32
8a+4b+2c+47 = -13
27a+9b+3c+47 = 16

f(x) = (52x^3 - 201x^2 + 104x + 47)/3

Answer 2

The sum of the coefficients of any polynomial $f(x)$ equals $f(1)$, since letting $x = 1$ in any term $cx^n$ gives us the coefficient $c$. That is, if
\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0,\]
then
\begin{align*}
f(1) &= a_n\cdot 1^n + a_{n-1}\cdot1^{n-1} + \cdots + a_1\cdot 1 + a_0\\
&=a_n+a_{n-1} + \cdots + a_1 + a_0.
\end{align*}
So, the sum of the coefficients of $f$ is $f(1) = \boxed{32}$.