Let A be a 6x6 matrix with characteristic polynomial x^3(x-1)(x+2)(x+4)

A) Prove that the rank of A is 3, 4 or 5.
B) If the rank of A is 5 is A diagonalizable?

A) I understand why rank must be less than 6 because  A is clearly not signular since it has 0 as an eigenvalue, however im having trouble actually proving that the rank must be 3, 4 or 5.

B) I think i may have this part. my thought is since Rank = 5 Dim of the kernal must be 1. Which means that the geometric multiplicity of the 0 eigenvalue is 1 where as its algebreic multiplicity is 3 and since they are not equal it is not diagonalizable. If i understand right, it would be diagonalizable if A was of rank 3 , correct?

Thanks for the help!