grandes-ecoles 2015 Q4

grandes-ecoles · France · x-ens-maths1__mp Roots of polynomials Polynomial evaluation, interpolation, and remainder

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We define the polynomials $$Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) \quad \text { and } \quad \forall j \in \{ 1 , \ldots , n \} , \quad P _ { j } = \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } .$$ (a) Show that the family $\left( Q _ { 0 } , P _ { 1 } , P _ { 2 } , \ldots , P _ { n } \right)$ is a basis of $\mathbb { R } _ { n } [ X ]$.
(b) Let $j \in \{ 1 , \ldots , n \}$. Verify that $( - 1 ) ^ { j - 1 } P _ { j } \left( \mu _ { j } \right) > 0$.

Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We define the polynomials
$$Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) \quad \text { and } \quad \forall j \in \{ 1 , \ldots , n \} , \quad P _ { j } = \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } .$$
(a) Show that the family $\left( Q _ { 0 } , P _ { 1 } , P _ { 2 } , \ldots , P _ { n } \right)$ is a basis of $\mathbb { R } _ { n } [ X ]$.\\
(b) Let $j \in \{ 1 , \ldots , n \}$. Verify that $( - 1 ) ^ { j - 1 } P _ { j } \left( \mu _ { j } \right) > 0$.

Paper Questions

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