How come a matrix not diagonalizable if the geometric multiplicity is less than the algebraic multiplicity?

Question

I understand that the algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. I also understand that the geometric multiplicity is the dimension of the eigenspace corresponding to some eigenvalue. I do not understand why having an eigenvalue appear m+1 times in the characteristic polynomial while having a dimension of m means that it is not diagonalizable. Is these something I am missing in my understanding of one of these concepts? It is not clear to me how they are connected.