The secret company project has been unveiled: a series of x springs which follow polynomial laws of force–displacement. as several constants are necessary to characterize these springs, you have been asked to evaluate the experimental data and provide the necessary calculations.

the x spring displacement is described by a polynomial extension of hooke's law,

f = -k x - q x^2 - r x^3

where k, q,r are the new x spring constants (note the order of coefficients and the again, the spring constants may be determined from the experimental data as the coefficients of the polynomial best fitting the displacement and force data. (keep the signs and order of coefficients and powers in mind! )

the data are recorded in the format [ m x ]; that is, the mass in the first column and the displacement observed in the second. units are unspecified in the experimental record.

quadratic x spring

10 2.152
20 8.880
40 38.53
80 137.7
160 528.3
320 2078.
640 8255.

cubic x spring

80 16.63
160 108.7
320 778.7
640 5989.

compose a function `fit_x_spring` which accepts an array of data in this format and returns the corresponding spring constants k, q,r. it should work the same for all orders of the polynomial behavior of the hyperelastic x spring.