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I am trying to solve following :

$$a(n)=2\cdot a(n-1)+5\cdot a(n-2)+6\cdot a(n-3)$$ with the initial conditions given by $a(0)=3,a(1)=2,a(2)=14$.

So first of all, I want to mark that there exists a theorem which states the following:

Let $P(x)=u_n\cdot x^n+u_{n-1}\cdot x^{n-1}+....+u_1\cdot x+u_0$ be a polynomial with real coefficients, and $u_n\neq 0$. If $P(x)$ has rational roots, they are of the form $\pm p/q$ where $p|u_0$ and $q|u_n$.

So first what I have tried is to construct characteristic equation or $$r^3-2\cdot r^2-5\cdot r-6=0$$ Now I have a problem to solve cubic equation also from the theorem I need a few help to understand it well.

Please help me. Thanks a lot.

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It looks to me you can't avoid solving the cubic if you want an explicit solution. Note that the cubic has one real and two conjugate complex roots. –  J. M. Sep 11 '11 at 11:36
    
Where is this question from? I don't think there are rational roots to this equation. –  Gadi A Sep 11 '11 at 11:37
1  
The lazy way: W|A –  Simon Sep 11 '11 at 11:45
    
it is good Simon just wanted to see how it can be solved by some algebraic methods thanks –  dato datuashvili Sep 11 '11 at 11:52
3  
This should be of interest... –  J. M. Sep 11 '11 at 12:04
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