The charachteristic polynomial of two similar matrices are the same. Thus, if $\displaystyle A$ has the charachteristic polynomial $\displaystyle X^3 - 4X^2 + 5X - 2$ then $\displaystyle A^3 - 4A^2 + 5A - 2I = \bold{0}$ by the
Cayley-Hamilton theorem. Of course, this theorem is very advanced and you probably never seen it before. Therefore, there is a weaker version for this theorem which states that a diagnolizable matrix satisfies its charachteristic polynomial. Since $\displaystyle A$ is a symettric matrix it means it is diagnolizable and the rest follows.
Thus, you need to find $\displaystyle \left[ \begin{array}{ccc}a&0&0\\0&b&0\\0&0&c \end{array} \right]$ which solve this polynomial equation.