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!
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!