grandes-ecoles 2014 QIII.B.3

grandes-ecoles · France · centrale-maths2__mp Roots of polynomials Factored form and root structure from polynomial identities

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$.
Let $P$ be a complex polynomial of degree 2. Justify the existence and uniqueness of $U \in G$ and $\alpha \in \mathbb{C}$ such that $U \circ P \circ U^{-1} = P_\alpha$. Determine these two elements when $P = T_2$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $G$ the set of complex polynomials of degree 1, and the inverse of $U \in G$ under composition is denoted $U^{-1}$.

Let $P$ be a complex polynomial of degree 2. Justify the existence and uniqueness of $U \in G$ and $\alpha \in \mathbb{C}$ such that $U \circ P \circ U^{-1} = P_\alpha$. Determine these two elements when $P = T_2$.

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.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C.1 QIII.C.2 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4