I'm trying to solve some exercises proposed by ENS of Paris. In particular the last one (which can be seen at http://www.ens.fr/IMG/file/SI2015/Sujets%20SCiences/Math2-version%20anglaise.pdf).
Since there are no solutions, I have a doubt!
I'm referring to second exercise.
1) In order to find $a_0$ I transformed the numerator as follows:
$(x-1)\cdot \dots \cdot (x-n) = (x+2-3)\cdot \dots \cdot (x-n) = (x+2)(x-2)\cdot \dots \cdot (x-n) + \dots$, where $\dots$ means the rest of the multiplication by $-3$. Then:
$=(x+2)(x+3-5)\cdot \dots \cdot (x-n) + \dots = (x+2)(x+3)\cdot \dots \cdot (x-n) + \dots$ and so on.
Then $a_0$ must be equal to $1$. Right?
For the second point I computed the integral of $T\cdot x^j$ for $1\leq j \leq n$ because if $T$ is orthogonal to the all generators, then it will be orthogonal to the whole space $F$. Then:
$\left< T, x^j \right> = \int_0^1\left( T(x) x^j \right)dx = \dots = \frac{a_n}{n+j+1} + \dots + \frac{a_0}{j+1}$ and this is zero because for $x = j$ then $S(X)= 0$. Right?
May I have a hint for the (3)?
Thank you very much!