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Vojtěch Jarník IMC
2022 VJIMC
1
polynomial has no real roots
polynomial has no real roots
Source: VJIMC 2022 1.1
April 11, 2022
algebra
polynomial
Problem Statement
Assume that a real polynomial
P
(
x
)
P(x)
P
(
x
)
has no real roots. Prove that the polynomial
Q
(
x
)
=
P
(
x
)
+
P
′
′
(
x
)
2
!
+
P
(
4
)
(
x
)
4
!
+
…
Q(x)=P(x)+\frac{P''(x)}{2!}+\frac{P^{(4)}(x)}{4!}+\ldots
Q
(
x
)
=
P
(
x
)
+
2
!
P
′′
(
x
)
+
4
!
P
(
4
)
(
x
)
+
…
also has no real roots.
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