Using FFT to calculate derivatives of polynomial
I think you would help yourself out to review the fundamentals of the Fourier Transform and the FFT, but I'll try to point you in the right direction.

First, i dont understand what is the ξ, when i learned FFT as much as i poorly understand it, we compute it at different complex roots.

\$\xi\$ is nothing more than another symbol for the frequency in cycles/second, which is more often labeled \$\nu\$. You can arrive at that formula quite trivially simply by starting with the expression for the inverse Fourier Transform, and differentiating with respect to time.

Now, i take the FFT of P(x) and get:


You have not taken the FFT of your polynomial. You have taken the FFT of the coefficients of your polynomial, which is nothing useful to you.
You have to first set up an array to represent some range of \$t\$, and then create an array \$f\$ which is the polynomial evaluated at each of those points \$t\$. Then, you can perform the fft on that array \$f\$. Let's say you call the resulting array (the Fourier Transform) \$\hat{f}\$. This will be an array of complex numbers. Each element of that array is the Discrete Fourier Transform of \$f\$ evaluated at a particular frequency. To know what frequency that is, you have to know that from one point in this array to the next, the frequency increases by \$\Delta\xi = \frac{1}{N\Delta t}\$, where \$N\$ is the number of points in your original array \$t\$, and \$\Delta t\$ is the spacing between adjacent points in your original array \$t\$.
Here you have to make sure you understand the convention of your fft package to know what the first element of the array is. Most likely it is the zero-frequency point.
So next to perform the operation of multiplying by \$2\pi i\xi\$, you need to take each element of the Fourier Transform array \$\hat{f}\$ and multiply it by \$2\pi i\xi\$ for that point, where \$\xi\$ is the frequency corresponding to that element in the Fourier Transform. This is where you go back and use the fact that the first point is (probably...you should verify ) \$\xi=0\$, and each subsequent point corresponds to an increase in frequency by \$\Delta\xi = \frac{1}{N\Delta t}\$.
Except....there is an unfortunate implementation complication at this stage that requires a little more understanding of how the FFT gets calculated. It is a bit much to go into all the theory here, but the result is that you need to 'fftshift' the frequencies back so that your frequency values run from \$-\frac{1}{2\Delta t}\$ to \$\frac{1}{2\Delta t}\$ rather than from 0 to \$\frac{1}{\Delta t}\$...in this form, you can multiply the array by the \$2\pi i\xi\$ array, which also must be offset so that the zero-frequency point occupies the same element in the array as does the zero-frequency point in the shifted fft array. I state all this for completeness, and I am sure this is a bit daunting if you are just starting off with working with FFT routines.
If you take this resulting array, fftshift it back, and apply an Inverse FFT on it, you will get an array which is the derivative of your polynomial. Due to computational issues, you will find that the number is complex, but if you have done everything correctly, the imaginary part will be zero to near floating point precision, and so the result is actually real. If it does not come out this way, you have made one or more mistakes.