If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Question

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if we write the coefficients of $f$ as $\dfrac{a_i}{b_i}, i = 0,.., n$, then we can multiply both sides of $f(x) = 0$ by $b = \prod_{i=0}^n b_i$ to achieve $f'(x) = 0$ where the coefficients of $f'$ can be viewed as elements of $A$. How do we get this to be $f(cx)$ though?