What is a simple root? differential equations

I know (x-c)^2 is a double root, but what is a simple root?

Also in the method of undetermined coefficients in differential equations, what does it mean to say that r is a simple root of the associated auxiliary equation?

simple example would be helpful

Thanks alot!

Thanks

A root of a polynomial is a number R, such that (x-R) divides the polynomial with no remainder (i.e., that (x-R) is a factor of the polynomial. If the polynomial can only be divided by (x-R) once, then R is a simple root. If the polynomial can be divided more than once (say, "n" times), then (x-R)^n is a factor, and the root is a "multiple root" (a root of multiplicity n).

Another way to think of this is in terms of the factorization of a polynomial:

P(x) = (x-A)*(x-B)*((x-C)^2)*((x-D)^5)

In this example, A and B are simple roots, while C and D are multiple roots, with multiplicity 2 and 5, respectively.

If R is a simple root to the auxiliary equation (aka characteristic equation), then exp(R*x) is a solution to the differential equation. If R is a multiple root of multiplicity n, then exp(R*x), x*exp(R*x), (x^2)*exp(R*x)...(x^(n-1))*exp(R*X) are all solutions to the equation.