Chapter 2, Question 12 from Axler's "Linear Algebra Done Right":
Suppose that $p_{0}, p_{1}, \ldots , p_{m}$ are polynomials in $P_{m}(F)$ such that $p_{j}(2) = 0$ for each $j$. Prove that $(p_{0}, p_{1}, \ldots , p_{m})$ is not linearly independent in $P_{m}(F)$.
What I have worked out so far:
Since $p_{j}(2) = 0$ for all $j$, $x^{0} = 1 \notin$ span$(p_{0}, p_{1}, \ldots , p_{m})$. That is , $(p_{0}, p_{1}, \ldots , p_{m})$ does not span $P_{m}(F)$.
Thanks.