Russia
Factorize:
@^
\begin{aligned}
p^3 + q^3 + p + q
\end{aligned}
@^
Answer:
^@ ( p + q ) ( p^2 - pq + q^2 + 1 ) ^@
- We know that @^ (a^3 + b^3) = (a + b) (a^2 - ab + b^2) @^ Thus @^ \begin{aligned} p^3 + q^3 + p + q = \space& ( p^3 + q^3 ) + ( p + q ) \\ = \space& ( p + q ) ( p^2 - pq + q^2) + ( p + q ) \\ = \space& ( p + q ) ( p^2 - pq + q^2 + 1 ) \end{aligned} @^
- Hence, ^@ p^3 + q^3 + p + q = ( p + q ) ( p^2 - pq + q^2 + 1 ). ^@
