grandes-ecoles 2022 Q8

grandes-ecoles · France · centrale-maths2__pc Factor & Remainder Theorem Polynomial Degree and Structural Properties

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is a polynomial of degree $n$. Explicitly state the leading coefficient of $T _ { n }$.

Let $n \in \mathbb{N}^*$ and
$$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$
Show that $T _ { n }$ is a polynomial of degree $n$. Explicitly state the leading coefficient of $T _ { n }$.

Paper Questions

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