grandes-ecoles 2019 Q20

grandes-ecoles · France · x-ens-maths1__mp Roots of polynomials Reciprocal and antireciprocal polynomial properties

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ Show that if $x \in \mathbb { C }$ is a root of $P _ { n }$, then $\frac { 1 } { x }$ is also a root of $P _ { n }$, with the same multiplicity.

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by
$$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$
Show that if $x \in \mathbb { C }$ is a root of $P _ { n }$, then $\frac { 1 } { x }$ is also a root of $P _ { n }$, with the same multiplicity.

Paper Questions

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