Efficient way to determine eigenvalues of a matrix (when one of them is known)

Hi all! I think the title says it all: once I have been given an exam question that gave us one of the eigenvalues of a matrix and asked us to calculate the other ones. However, one restriction was given: we could use any method we'd like, EXCEPT calculating the characteristic polynomial and then deflacionating it.

The context was a Numerical Analysis exam, and our assumed basis were methods to solve linear systems, including direct methods such as Gauss's, Cholesky Decomposition, LU Decomposition, LDL Decomposition and iteractive-stationary methods, such as Jacobi's and Gauss-Seidel's.

So, I'm asking for an approach. I did some research and found the Power Iteration algorithm, but I'm not sure we had seen something like this in class. So, for the sake of exercise, I'm looking for answers which don't involve this one.

Thanks in advance!