Find the product:

(x – 2y – z)(x2 + 4y2 + z2 + 2xy + zx + 2yz)

We have,


a3 (b – c)3 + b3 (c – a)3 + c3 (a – b)3


We know that,


a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)]


Also,


If, (x + y + z) = 0


Then (x3 + y3 + z3) = 3xyz


Using this, we get


= [x + (-2y) + (-z)] [(x)2 + (-2y)2 + (-z)2 – (x) (-2y) – (-2y) (-z) – (-z) (x)


= (a + b + c) (a2 + b2 + c2 – ab – bc – ca)


= a3 + b3 + c3 – 3abc


Where,


x = a, b = -2y and c = -z


(x – 2y – z) (x2 + 4y2 + z2 + 2xy + zx – 2yz)


= (x)3 + (-2y)3 + (-z)3 – 3 (x) (-2y) (-z)


= x3 – 8y3 – z3 – 6xyz


17