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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 $f$?

 Mar 21, 2020
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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  x3    -67x2   + 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 ) 

 Mar 21, 2020
edited by ElectricPavlov  Mar 21, 2020
edited by ElectricPavlov  Mar 21, 2020

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