Characterization of polynomials which have no zeros in the unit interval In 1D, it is straightforward to create real-valued coefficients polynomials which have no roots in $[0, 1]$: Pick a degree $n$ and some parameters $a_i$ outside of $[0, 1]$, and expand
$$
p(x) = b \prod_{i=1}^n (x - a_i).
$$
This isn't fully general though: Polynomials like $x^2+1$ are missed.
Is there a canonical form of real-valued polynomials with no zeros in $[0, 1]$?

The fully general form is
$$
\alpha \prod_i L_i(x) \prod_j Q_j(x)
$$
where $L$ is a monic linear polynomial without roots in $[0,1]$ and $Q_j$ is a monic quadratic polynomial without real roots. Each product may be empty. In other words,
$$
\alpha \prod_i (x-a_i) \prod_j (x^2+b_jx+c_j)
$$
where $\alpha, a_i, b_j,c_j \in \mathbb R$ with $\alpha\ne0$, $a_i \notin [0,1]$ and $b_j^2-4c_j <0$.
