grandes-ecoles 2017 Q25

grandes-ecoles · France · x-ens-maths__psi Roots of polynomials Location and bounds on roots

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The polynomial $K(X)$ is defined as in question 23.
Deduce that the roots of $K$ are all real and belong to the interval $]0, 4[$.

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product
$$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$
The polynomial $K(X)$ is defined as in question 23.

Deduce that the roots of $K$ are all real and belong to the interval $]0, 4[$.

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