A polynomial has only real roots and all coefficients $\pm 1$. Prove the degree $<4$.

Question

Let $P(x)$ be a polynomial with only real roots and all coefficients equal to $\pm 1$. Prove that the degree of the polynomial is less than 4.

This is practice for Putnam, but I am not certain where to begin. I know I need to use inequalities. Could this be a roots of unity problem?

Answer 1

Let $P(x)=x^n+a_1x^{n-1}+...+a_n$ where all $a_i'$ s are $1$ or $-1$ and $x_1,x_2...x_n$ are all real roots of $P.$ By Viete's formulas $|x_1+x_2+...+x_n|=1$ and $x_1^2+x_2^2+...+x_n^2=(x_1+x_2+...x_n)^2-2\sum_{i<j}x_ix_j=1-2a_{2}=3.$ Now we can estimate $3=x_1^2+x_2^2+...x_n^2\ge n\sqrt[n]{x_1^2x_2^2...x_n^2}=n$ so, $n\le 3$ and the result follows.