Let $f$ be in $ L_{\text {loc }}^1$, and suppose that there exists a polynomial $p$ such that $f / p$ is integrable on $\mathbf{R}$. Show that $$ T_f(\varphi):=\int f \varphi d x, \quad \varphi \in \mathcal{S} $$ is a tempered distribution.
My attempt: Suppose that $p$ is a polynomial of degree $k$
$| \int_{\mathbb{R}} f(x) \varphi(x) dx | \leq \int_{\mathbb{R}} |f(x)/p(x)| |p(x)\varphi(x)| dx \leq C \mathcal{N}_{0, k} \int_{\mathbb{R}} |p(x) \phi(x)|dx \leq + \infty $ by hypothesis.
Your proof should be:
\begin{align*} \left|\int_{\mathbb{R}}f(x)\varphi(x)dx\right| &\leqslant \int_{\mathbb{R}}\left|\frac{f(x)}{p(x)}\right||p(x)\varphi(x)|dx \\&\leqslant \sup|p(x)\varphi(x)|\int_{\mathbb{R}}\left|\frac{f(x)}{p(x)}\right|dx \\&=C\sup|p(x)\varphi(x)| \end{align*}
This means that $T_{f}(\varphi)$ is tempered distribution.