Trace of $A^4$ in terms of Tr$(A^3)$, Tr$(A^2)$ and Tr$(A)$ for a 3x3 matrix

Question

If $A$ is a 3x3 matrix, can Tr$(A^4)$ be related to Tr$(A^3)$, Tr$(A^2)$, and Tr$(A)$, where Tr is the trace of the matrix?

If so, how can we prove it? A similar question has been answered in Expressing the trace of $A^2$ by trace of $A$.