Conservative polynomials · Almost Obvious

Let \(P\) be a real polynomial that takes both positive and negative values. Prove that \(\lvert P\rvert\) is not a polynomial.

Hint

If \(Q=\lvert P\rvert\) were a polynomial, what would squaring tell you?

Solution

Suppose \(Q=\lvert P\rvert\) is a polynomial. Since \(Q^2=P^2\) at every real number, we have the polynomial identity

\[(Q-P)(Q+P)=0.\]

A product of two nonzero polynomials is nonzero, so \(Q=P\) or \(Q=-P\). Because \(Q\geq0\) everywhere, either choice forces \(P\) to have only one sign—a contradiction.