I have been given a 3x3 complex matrix and been told to find the determinant of (A-[lamba]I), which I have calculated (and double and triple checked) to be the following:
-lambda^3 + w^2[lambda] + u^2[lambda] + v^2[lambda] + 2uvw
where u, v, and w are complex numbers in the matrix A. The question is this: How can I use the fact that C (the field of complex numbers) is algebraically closed to conclude that det(A-[lambda]I) = (lambda - alpha)(lambda - beta)(lambda - gamma) for some complex numbers alpha, beta, gamma?