To use the characteristic equation of a recurrence relation to get a closed form, we must find the roots of the characteristic equation.

According to the Lecture 13 slides:

Find the roots of the characteristic equation:

(r − α_1)(r − α_2)· · ·(r − α_k ) = 0 where all the α_i are distinct.

How are we supposed to proceed if we have some/all non-distinct roots? For instance, the 2nd example on the next slide has the characteristic equation: (r+1)^3