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?
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
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 $f(1) = \boxed{32}$.
To find the sum of the coefficients of the polynomial f, we need to determine the degree of the polynomial and then find each coefficient.
Given that we have four points (0, 47), (1, 32), (2, -13), and (3, 16), we can use these points to construct a system of equations.
We can set up the following system of equations:
f(0) = a(0)^3 + b(0)^2 + c(0) + d = 47
f(1) = a(1)^3 + b(1)^2 + c(1) + d = 32
f(2) = a(2)^3 + b(2)^2 + c(2) + d = -13
f(3) = a(3)^3 + b(3)^2 + c(3) + d = 16
where a, b, c, and d are the coefficients of the polynomial f.
Substituting the values we know, we get:
a(0) + c = 47
a(1)^3 + b(1)^2 + c + d = 32
a(2)^3 + b(2)^2 + 2c + d = -13
a(3)^3 + b(3)^2 + 3c + d = 16
To solve this system of equations, we can use a variety of methods (e.g., substitution, elimination, or matrices).
One possible approach is to substitute for c in terms of a, using the equation a(0) + c = 47.
Since f(0) is specified to be 47, we can determine that a(0) = 0 and c = 47.
Substituting these values back into the equations, we get:
a(1)^3 + b(1)^2 + 47 + d = 32
a(2)^3 + b(2)^2 + 94 + d = -13
a(3)^3 + b(3)^2 + 141 + d = 16
Now, we have three equations with three unknowns: a, b, and d.
Solving this system of equations will give us the values of a, b, and d, which are the coefficients of the polynomial f.
Once we find the values of a, b, c, and d, we can calculate the sum of the coefficients by adding all the coefficients together.
Please note that solving this system of equations can be tedious and time-consuming, especially for higher degrees of the polynomial. In this case, the given points specify a polynomial of degree 3, so you will find the values of the coefficients of a cubic polynomial.