Could someone walk me through finding the roots of higher order polynomials? for example r^4-2r^2-1=0? I'm working on higher order DE's and the roots of the characteristic equation keeps tripping me up. I'm wondering if there's some method from algebra I've completely forgotten or something.

Question

mathmate · Answer

In this case, you can substitute u=r^2 to get a quadratic, and solve accordingly.

anonymous · Answer

how would the -1 be handled? i've never heard of using substitution to solve something like this. sorry I really dont know how i've managed to get this far with such a crappy algebra background haha

mathmate · Answer

u=r^2
r^4-2r^2-1=0 => u^2-2u-1=0
Solve by the quadratic formula
u=(2+/- sqrt(4+4))/2
=1 +/- sqrt(2)
r=+/- sqrt(1+sqrt(2)) or +/- sqrt(1-sqrt(2))
The first two are in R, so e functions are solutions.
The last two solutions are in C, so you'd end up with some sine/cosine functions.

anonymous · Answer

ohh gotcha thanks I appreciate it!

mathmate · Answer

You're welcome!

anonymous · Answer

but does this work for imaginary numbers?

mathmate · Answer

You mean the substitution?  Yes.