A 0-1-2 polynomial is a univariate polynomial where all coefficients are either $0$, $1$, or $2$. Is it true that if two real polynomials $P(x), Q(x)$, have their product equal to a 0-1-2 polynomial (e.g., $1+x+x^5+ 2 x^{42}$), and their coefficients are assumed to be non-negative, and are both monic, then their coefficients are also 0's, 1's and 2's ?
The special case with no $2's$ is posted here. The question here is a generalization of that question, where coefficients are 0, 1, or 2, instead of 0 or 1.
The conjecture is false. Counterexample:
$$ x^3 + x^2 + 2x + 1 = (x+a)(x^2+bx+c)$$ with $a \approx 0.5698$, $b \approx 0.4302$, $c\approx 1.755$.