Problem: Let $P(x)$ and $Q(x)$ be arbitrary polynomials with real coefficients,
with $P \neq 0$, and let $d = \deg P$.
Prove that there exist polynomials $A(x)$ and $B(x)$,
not both zero, such that $\max \{ \deg A, \deg B \} \le d/2$
and $P(x) \mid A(x) + Q(x) \cdot B(x)$.

Solution: Let $V$ be the vector space of real polynomials with degree at most $d/2$.
Consider maps of linear spaces
\begin{align*}
  V^{\oplus 2} &\to \RR[x] / (P(x)) \\
  \text{by} \qquad (A,B) &\mapsto A+QB \pmod P.
\end{align*}
The domain has dimension
\[ 2 \left( \left\lfloor d/2 \right\rfloor + 1 \right) \]
while the codomain has dimension $d$.
For dimension reasons it has nontrivial kernel.