AoPS Wiki talk:Problem of the Day/July 1, 2011
Problem
AoPSWiki:Problem of the Day/July 1, 2011
Solution 1
To factor , we should try to find a way to create a quadratic in disguise. There, in fact, is a way!
Expand and separately:
We notice that there is a in both of these terms! Treat as a single quantity, .
Our original quantity was .
We can factor the original polynomial as .
can be factored as .
Our final factorization is .
Solution 2
Let . Then and (since ). Therefore I have: where is a quadratic. Simplifying yields as before, which is irreducible as .