How do you solve ?

It's a characteristic polynomial. Is it possible to solve it from here or must I solve them without opening too many brackets?

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- Oct 25th 2009, 09:43 PMgeftSolving cubic polynomials
How do you solve ?

It's a characteristic polynomial. Is it possible to solve it from here or must I solve them without opening too many brackets? - Oct 25th 2009, 10:47 PMearboth
The LHS of the equation can't be factored. Therefore you can use either the Cardanic formula (see here: Cubic function - Wikipedia, the free encyclopedia) or an iterative method like Newton's method to get an approximate solution.

- Oct 25th 2009, 11:00 PMgeft
It shouldn't be that complicated since I wasn't taught that method. Here's the original equation.

- Oct 26th 2009, 03:40 AMHallsofIvy
You could try looking for "rational roots". The "rational root theorem" says that any rational roots of are of the form x= r/s where s is an integer factor of a, the leading coefficient, and s is an integer factor of c, the constant term.

Here, the leading coefficient is "3" which has factors 1, -1, 3, and -3. The constant term is "8" which has factors 1, -1, 2, -2, 4, -4, 8, and -8. The only possible rational roots are 1, -1, 2, -2, 4, -4, 8, -8, 1/3, -1/3,, 2/3, -2/3, 4/3, -4/3, 8/3, and -8/3. Trying each of those will either give you a root or show that this equation has no rational roots. If there are no rational roots there will not be any solution simpler than Cardano's. - Oct 26th 2009, 03:56 AMmr fantastic
- Oct 26th 2009, 06:54 AMgeft
You're right. Here's the matrix:

- Oct 26th 2009, 06:02 PMmr fantastic
- Oct 26th 2009, 09:39 PMgeft

???

- Oct 26th 2009, 11:43 PMmr fantastic
- Oct 26th 2009, 11:55 PMgeft
Wow, somehow I got it.