grandes-ecoles 2017 QIII.E.1

grandes-ecoles · France · centrale-maths2__psi Roots of polynomials Multiplicity and derivative analysis of roots
We still assume that $p$ is an integer greater than or equal to 2. Let $R \in \mathbb { C } [ X ]$ be a polynomial of degree greater than or equal to 1 with simple roots. Prove that the polynomial $R \left( X ^ { p } \right)$ has simple roots if and only if $R ( 0 ) \neq 0$. 
We still assume that $p$ is an integer greater than or equal to 2. Let $R \in \mathbb { C } [ X ]$ be a polynomial of degree greater than or equal to 1 with simple roots. Prove that the polynomial $R \left( X ^ { p } \right)$ has simple roots if and only if $R ( 0 ) \neq 0$.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.C QII.D.1 QII.D.2 QII.E.1 QII.E.2 QII.E.3 QIII.A QIII.B.1 QIII.B.2 QIII.C QIII.D.1 QIII.D.2 QIII.E.1 QIII.E.2 QIII.E.3 QIV.A.1 QIV.A.2 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2 QV.A QV.B.1 QV.B.2 QV.B.3 QV.B.4