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? Please explain.

With four values given, the polynomial could be a cubic:

 

f(x) = ax³ + bx² + cx + d

 

Replace x by 1 to get: 

f(1) = a + b + c + d. ie the sum of the coefficients.

 

But we are told that f(1) is 32, Hence this is the sum of the coefficients.

 

This reasoning holds even if the polynomial is not a cubic, as replacing x by 1 simply results in f(1) being the sum of the coefficients!