This is actually for my ODE class, but I'm stuck on a factorization type question.
r^5 - 3r^4 + 3r^3 - 3r^2 + 2r = 0
First thing I can do is factor out an r, leaving me with:
r ( r^4 - 3r^3 + 3r^2 - 3r^1 + 2) = 0.
One root is obviously 0. Cool. Now how do I factor the expression in the parentheses to find the roots of the equation? The method my book advises involves finding the factors of a0 (the coefficient of r^4 in this case) and an (2, in this case). So, I am left with +/- 1 and +/- 2. I tried plugging these into the equation and found that +1 and +2 were also roots.
BUT apparently there is a way to factor the above parenthetical expression into (r-1)(r-2)(r^2 + 1), in which case we would find the roots listed above, AS WELL AS +/- i. So, what is the method used to factor that equation that way? Since it seems the coefficient method was not completely sufficient.