Find the Product:
\[ (3x – 4y + 5z) \] \[ (9x^2 + 16y^2 + 25z^2 + 12xy – 15zx + 20yz) \]
Solution:
Using identity:
\[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc \]
Rewrite:
\[ 3x-4y+5z = 3x+5z-4y \]
So, \[ a=3x,\qquad b=5z,\qquad c=-4y \]
\[ (3x – 4y + 5z) \] \[ (9x^2 + 16y^2 + 25z^2 + 12xy – 15zx + 20yz) \]
\[ = (3x)^3+(5z)^3+(-4y)^3 -3(3x)(5z)(-4y) \]
\[ = 27x^3+125z^3-64y^3+180xyz \]