This is problem #15 on page 189 of Axler's book Linear Algebra Done Right.
Suppose V is a complex vector space. Suppose T is in L(V) is such that 5 and 6 are eigenvalues of T and that T has no other eigenvalues.
Prove that (T − 5I)^(n−1)*(T − 6I)^(n−1) = 0, where n = dimV.
A Complex Vector Space and its Eigenvalues are investigated.