$x=\mathrm{p}+\mathrm{q}, y=\mathrm{p}+$ $\omega q, z=p+\omega^{2} q$ হলে $x^{3}+y^{3}+z^{3}$=?

Solve: দেওয়া আছে, $x=\mathrm{p}+\mathrm{q}, y=\mathrm{p}+$ $\omega q, z=p+\omega^{2} q$

$\begin{aligned} & x^{3}+y^{3}+z^{3}=(p+q)^{3}+(p+\omega q)^{3} + & \left(p+\omega^{2} q\right)^{3} \\ = & \left.p^{3}+q^{3}+3 p^{2} q+3 p q^{2}+p^{3}+\omega^{3} q^{3}+3p^{2} q \omega\right) \\ + & \left.3 p \omega^{2} q^{2}+p^{3}+\omega^{6} q^{3}+3 p^{2} q \omega^{2}+3p q^{2} \omega^{4}\right) \\ = & p^{3}+q^{3}+3 p^{2} q+3 p q^{2}+p^{3}+q^{3}+3 p^{2} q \omega \\ & \left.+3 p \omega^{2} q^{2}+p^{3}+q^{3}+3 p^{2} q \omega^{2}+3 p q^{2} \omega\right) \\ = & 3\left(p^{3}+q^{3}\right)+3 p^{2} q\left(1+\omega+\omega^{2}\right)+ \\ \quad & 3 p q^{2}\left(1+\omega+\omega^{2}\right) \\ \therefore & x^{3}+y^{3}+z^{3}=3\left(p^{3}+q^{3}\right) \end{aligned}$