Mock AIME 6 2006-2007 Problems/Problem 7

Problem

Let $P_n(x)=1+x+x^2+\cdots+x^n$ and $Q_n(x)=P_1\cdot P_2\cdots P_n$ for all integers $n\ge 1$ . How many more distinct complex roots does $Q_{1004}$ have than $Q_{1003}$ ?

Solution

The roots of $P_n(x)$ will be in the form $x=e^{\frac{2\pi k}{n+1}}$ for $k=1,2,\cdots,n$ with the only real solution when $n$ is odd and $k=\frac{n+1}{2}$ and the rest are complex.

Therefore each $P_n(x)$ will have $n$ distinct complex roots when $n$ is even and $n-1$ distinct roots when $n$ is odd.

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

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Mock AIME 6 2006-2007 Problems/Problem 7

Problem

Solution