I have to find a polynomial with the following characteristics for a problem.
Find a polynomial $p(x)$ such that $$p(-1)=p'(-1)=p''(-1)=p(1)=p'(1)=p''(1)=0$$
I know and understand the process of how to find u(x), But since I don't know which
polynomial behave like this I'm stuck in the problem. I will be so thankfully if
somebody can help me to find this polynomial. The rest of the problem is so easy,
once I identify this polynomial.
If you have that $p(a)=0\Rightarrow p(X)=(X-a)q(X)$, a fact which comes from what it is knows as Bezout's little theorem (http://en.wikipedia.org/wiki/Polynomial_remainder_theorem), which can be proved using Euclid's algorithm.
If you have that $p(a)=p'(a)=0\Rightarrow p(X)=(X-a)^2r(X)$. Why? $p(X)(X-a)q(X)\Rightarrow p'(X)=q(X)+(X-a)q'(X)\Rightarrow q(a)=0\Rightarrow q(X)=(X-a)r(X)\Rightarrow p(X)=(X-a)^2r(X)$ and you can go on with the process. Therefore $(X-1)^3(X+1)^3$ check your problem.