Extruding a polynomial

I happened to wonder whether you could take any polynomial like for example:

and write it like this, as linear factors multiplied and summed:

As it turns out with this polynomial it was possible, I just set the two equal at 20 different points and solved for the 20 variables...

And the coefficients are:

I found it interesting that all of these coefficients are real numbers, I'm not sure if it's ever necessary to use imaginary numbers or not. I called it extruding the polynomial because it reminds me of the way steel can be shaped by squeezing it through an aperture, in this case a linear factor aperture :). 

You might wonder why I didn't try to make z:

Maple wasn't able to solve this one even with so many less variables...