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 $f$?

Guest Mar 21, 2020

#1**+1 **

If f(0) = 47 then 'the constant term = 47

ax^3 + bx^2 + cx +47

if f(1) = 32

a(1^3) + b(1^2) + c + 47 = 32

a + b + c + 47 = 32

If f(2) = -13

a (2)^3 + b (2)^2 + 2c +47 = -13

8a + 4b + 2c + 47 = -13

If f(3) = 16

27a + 9b + 3c + 47 = 16

Solving the system of equations in red (I used an online calculator) results in f(x) = 52/3 x^{3 }-67x^{2 }+ 104/3 x + 47

you can take it from here......

***** Edit *** I think I took the long way.... f(1) = 32 would give the answer ! D'Oh ! (IF you consider the constant 47 a 'coefficient'.....if not then the sum would be -15 ) **

ElectricPavlov Mar 21, 2020