grandes-ecoles 2017 Q10

grandes-ecoles · France · x-ens-maths1__mp Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity

Let $P , Q \in \mathbb { R } [ X ]$ be nonzero polynomials of respective degrees $p$ and $q$ strictly positive. Show that the linear map $L _ { P , Q }$ defined by $$\left\lvert \, \begin{array} { c c c } L _ { P , Q } : \quad \mathbb { R } _ { q - 1 } [ X ] \times \mathbb { R } _ { p - 1 } [ X ] & \rightarrow \quad \mathbb { R } _ { p + q - 1 } [ X ] \\ ( V , W ) & \mapsto V P + W Q \end{array} \right.$$ is an isomorphism if and only if $P$ and $Q$ are coprime in $\mathbb { R } [ X ]$.

Let $P , Q \in \mathbb { R } [ X ]$ be nonzero polynomials of respective degrees $p$ and $q$ strictly positive. Show that the linear map $L _ { P , Q }$ defined by
$$\left\lvert \, \begin{array} { c c c } L _ { P , Q } : \quad \mathbb { R } _ { q - 1 } [ X ] \times \mathbb { R } _ { p - 1 } [ X ] & \rightarrow \quad \mathbb { R } _ { p + q - 1 } [ X ] \\ ( V , W ) & \mapsto V P + W Q \end{array} \right.$$
is an isomorphism if and only if $P$ and $Q$ are coprime in $\mathbb { R } [ X ]$.

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