checking if matrix is diagonalizable

My textbook gives two theorems that are used to determine if a matrix is diagonilable

1)An n x n matrix is diagonalizable if and only if it has n linearly independent eigenvectors

2)If the roots of the characteristic polynomial of an n x n matrix A are all distinct then A is diagonizable

So if I have characteristic polynomial with two roots that are the same (e.g. $\displaystyle \lambda=1$ but it has n linearly independent eigenvectors, which one takes presidents? Or should the two always agree and this means I made a mistake?