Show that $x$ is algebraic over $F$ Let $x$ be an indeterminate over $\mathbb{Q}$, and let $F=\mathbb{Q}(7x^3/(5x-13))$. Note that $F \subseteq \mathbb{Q}(x)$. Show that $x$ is algebraic over $F$ by expressly giving a polynomial over $F$ that $x$ is a root of. My main issue is understanding what is meant by $\mathbb{Q}(7x^3/(5x-13))$ and how to find potential polynomials.

$\mathbf Q\Bigl(\dfrac{7x^3}{5x-13}\Bigr)$ simply denotes the set of rational functions in $u=\dfrac{7x^3}{5x-13}$. What else?

As to a polynomial equation satisfied by $x$ over $\mathbf Q(u)$, it's pretty simple: $$u=\frac{7x^3}{5x-13}\iff 7x^3=u(5x-13)\iff 7x^3-5ux+13u=0,$$

so $\;[\mathbf Q(x):\mathbf Q(u)]=3.$