Let $\mathbb{R}[t]_{\leq n}$ represent the vector space of polynomials (with coefficients in $\mathbb{R}$) whose degree is at most $n$. $\forall a \in \mathbb{R}$, let $U_a = \{P(t) \in \mathbb{R}[t] \mid P(a) = 0 \}$
1) Find a basis of $S = U_a \cap \mathbb{R}[t]_{\leq n}$ for all $a \in \mathbb{R}$
For all $P(t) \in S$, $ P(t) = \lambda_0 + \lambda_1t + \lambda_2 t² + \cdots + \lambda_nt^n$ for $\lambda_i \in \mathbb{R}, \forall i$. We also know that $P(a) = 0$ so we can write:
$\sum_{i=0}^{n}(\lambda_ia^i) = 0$
But beyond this, I'm not sure how to proceed..
2) Show that $(U_3+U_4) \cap \mathbb{R}[t]_{\le n} = \mathbb{R}[t]_{\le n}$.
I'm stumped. Wouldn't this mean that $\forall p \in \mathbb{R}[t]_{\leq n}$, p can be written $p=Q(t)(t-3)+Q'(t)(t-4)$ with $deg(Q)$, $deg(Q') < deg(P)$ ?