Finding a specific polynomial function, with conditions. 
Find a polynomial such that $f(x) = a + bx + cx^2 +dx^3 + ex^4$ such
that $f(1)=1, f(2) = -1, f(-1)=5, f(3) = -59, f(-2) = -29$

Any hints on how to approach this? I was thinking about plugging in the f value and then the result in order to build a matrix like this, but I am not sure if that is what I am supposed to do here.
$\displaystyle \begin{pmatrix}
a & b & c & d & e & 1\\
a & -b & -c & -d & -e & -1\\
a & -b & -c & -d & -e & 5\\
a & 3b & 3c & 3d & 3e & -59\\
a & -2b & -2c & -2d & -2e & -29
\end{pmatrix}$
does this make any sense?

$f(1)=1, f(2)=-1,f(-1)=5,f(3)=-59,f(-2)=-29$
Provided a linear $5\times 5$ sysytem of equation as $MV=U$
where $$M=\begin{pmatrix} 1 & 1 &1 &1 &1 \\ 1 & 2 & 4 & 8 & 16 \\ 1 & -1 & 1 &-1 & 1\\ 1 & 3 & 9 & 27 & 81 \\ 1 & -2 & 4 & -8 & 16 \end{pmatrix}, V=\begin{pmatrix} a \\ b\\ c\\ d\\ e \end{pmatrix}, U =\begin{pmatrix} 1 \\ 2 \\ -1 \\ 5 \\ -59 \\ -29 \end{pmatrix}.$$ Solving this equation by LinearSolve of Mathematica we get get $$V=\begin{pmatrix} 1 \\ -5 \\ 4 \\ 3 \\ -2 \end{pmatrix}$$
Finally, we get $f(x)=1-5x+4x^2+3x^3-2x^4.$
Mathe matica Command: LinearSolve[m,u]
