grandes-ecoles 2014 QI.A.4

grandes-ecoles · France · centrale-maths2__mp Roots of polynomials Location and bounds on roots

The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that, for every natural integer $n$, the polynomial $T_n$ is split over $\mathbb{R}$, with simple roots belonging to $]-1,1[$. Determine the roots of $T_n$.

The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.

Show that, for every natural integer $n$, the polynomial $T_n$ is split over $\mathbb{R}$, with simple roots belonging to $]-1,1[$. Determine the roots of $T_n$.

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