10x^4y^3-5x^3y^2+20x^2y10x4y3−5x3y2+20x2y
=5x^2y(2x^2y^2-xy+4)=5x2y(2x2y2−xy+4)
=5x^2y(2(xy)^2-(xy)+4)=5x2y(2(xy)2−(xy)+4)
This is as far as we can go with real coefficients.
The roots of 2t^2-t+4 = 02t2−t+4=0 are given by the quadratic formula as:
t = (1+-sqrt(1^2-(4xx2xx4)))/(2*2)t=1±√12−(4×2×4)2⋅2
=(1+-sqrt(-31))/4=1±√−314
=1/4+-i sqrt(31)/4=14±i√314
So if we allow complex coefficients we get:
2(xy)^2-(xy)+42(xy)2−(xy)+4
= 2(xy - 1/4 -i sqrt(31)/4)(xy - 1/4 + i sqrt(31)/4)=2(xy−14−i√314)(xy−14+i√314)
Hence:
10x^4y^3-5x^3y^2+20x^2y10x4y3−5x3y2+20x2y
=10x^2y(xy - 1/4 -i sqrt(31)/4)(xy - 1/4 + i sqrt(31)/4)=10x2y(xy−14−i√314)(xy−14+i√314)
