Problem
Prove or disprove: there exists a basis $p_0, p_1, p_2, p_3$ of $\mathcal{P}_3(\mathbb{F})$ such that none of the polynomials $p_0, p_1, p_3$ has degree 2.
Solution
The list $z^2, z^3, z, 1$ is a basis for $\mathcal{P}_3(\mathbb{F})$ by example 2.28 (g), and therefore, by P2A7, so is the list $5z^2 - 4z^3, z^3, z, 1$ in which none of the polynomials has degree 2.