Find the Product:
\[ (4x – 3y + 2z) \] \[ (16x^2 + 9y^2 + 4z^2 + 12xy + 6yz – 8zx) \]
Solution:
Using identity:
\[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \]
Rewrite:
\[ 4x-3y+2z = 4x+2z-3y \]
So, \[ a=4x,\qquad b=2z,\qquad c=-3y \]
\[ (4x – 3y + 2z) \] \[ (16x^2 + 9y^2 + 4z^2 + 12xy + 6yz – 8zx) \]
\[ = (4x)^3+(2z)^3+(-3y)^3 -3(4x)(2z)(-3y) \]
\[ = 64x^3+8z^3-27y^3+72xyz \]