how do you factor a 3rd or 4th order polynomial. ex. D^3+7D^2+14D+8=0

Question

vishweshshrimali5 · Answer

well it is better if u suppose x ^ 4 = y^2 . That will make it a quadratic equation and then further solve it by putting x^2 = y

anonymous · Answer

ok, it's just I was wondering if there was a failsafe way of solving higher order polynomias. I've been out of Algebra for a while and need this for solving differential equations. I can't seem to figure out where they get some of the roots from

anonymous · Answer

would synthetic division work? idk this is really bugging me, and if you were to use synthetic division how are you to determine what the first (D+/-Co) would be

phi · Answer

4th order is the highest order we have formulas for. But they are very ugly. Here's 3rd order (cubics) http://en.wikipedia.org/wiki/Cubic_function#General_formula_of_roots

jamesj · Answer

Phi is exactly right.  

usually the best thing is to experiment and see if you find a solution or estimate a solution by find numbers a and b such that p(a) < 0 < p(b) or vice versa.

In your case with D^3+7D^2+14D+8
you can see that D = -1 is a solution.  

Now factor out D+1 and you're in good shape as you will now have a quadratic.

anonymous · Answer

once i find (D+1) i see where to go from there. i just dont see how you can tell from D^3+7D^2+14D+8, that D+1 is a solution, rather D=-1

jamesj · Answer

I could not see that D+1 is a factor of p(D) = D^3+7D^2+14D+8

But it's not hard to see D = -1 is a solution: plug it in to p(D) and show that P(D=-1) = 0.  

D = -1 was the second number I tried after D = 1, which didn't work.

The point now is that because D = -1 is a root of the p(D), D+1 is a factor of the polynominal.  Do the long division and find a new polynominal q(D) such that

p(D) = (D+1)q(D)

This the real reason you really learnt long division in high school. :-) Not to find things like 3486/42, but to do this sort of long division.

anonymous · Answer

ok, I see. thanks for the help. I think it's funny that reason Im having trouble on my DiffEQ hw is because of factoring and not the actual DEQ material. thanks