Operations are suggested by the form of factors required.
Multiply C1 by x and hence divide by x and then apply
C1+(yC2+zC3), we get
Δ=1x⎡⎢
⎢⎣a(x2+y2+z2)ay+bxcx+azb(x2+y2+z2)by−cz−axbz+cyc(x2+y2+z2)bz+cycz−ax−by⎤⎥
⎥⎦
Take ∑x2 from C1 and then again multiply R1 by a and divide Δ by a Now operate
R1+(bR2+cR3)
Δ=1ax(x2+y2+z2)
⎡⎢
⎢
⎢⎣a2+b2+c2y(a2+b2+c2)z(a2+b2+c2)bby−cz−axbz+cycbz+cycz−ax−by⎤⎥
⎥
⎥⎦
=1ax(x2+y2+z2)(a2+b2+c2)
×⎡⎢⎣1yzbby−cz−axbz+cycbz+cycz−ax−by⎤⎥⎦
Now make two zeros by applying R2−bR1 and R3−CR1
∴Δ=⎡⎢⎣1yz0−(cz−ax)cy0bz−(ax−by)⎤⎥⎦
=(cz+ax)(ax+by)−bcyz
=ax(ax+by+cz) ∴Δ=∑x2.∑a2.∑ax