Characteristic Polynomial from a matrix with an unknown...

Hi, for the second time today, still linear algebra...

Okay, if I have the matrix,

A = [2a a 0]

--- [ a a 0]

--- [ 0 0 a]

and I want to find the eigenvalues of the matrix, I know I need to set up the characteristic polynomial, which I find to be...

(2a-t)(a-t)^2 - a^2(a-t)

I have made various adjustments, but I still haven't found the roots of the polynomial.

What I have come up with is either...

-t^3 + 4at^2 - 4a^2t - a^3

or

(a-t)(t^2 - 3at + a^2)

Okay, so... No matter which equation I look at, I still can't find the roots.

I can see from the last one -- that one root should be t = a? But division comes up with an illegal answer.

So, if anyone has any input, it is very welcome.

Thanks.

Simon DK.