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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2019 Turkey Team SeIection Test
5
5
Part of
2019 Turkey Team SeIection Test
Problems
(1)
Polynomial with real roots
Source: Turkey TST 2019 Day 2 P5
3/26/2019
P
(
x
)
P(x)
P
(
x
)
is a nonconstant polynomial with real coefficients and its all roots are real numbers. If there exist a
Q
(
x
)
Q(x)
Q
(
x
)
polynomial with real coefficients that holds the equality for all
x
x
x
real numbers
(
P
(
x
)
)
2
=
P
(
Q
(
x
)
)
(P(x))^{2}=P(Q(x))
(
P
(
x
)
)
2
=
P
(
Q
(
x
))
, then prove that all the roots of
P
(
x
)
P(x)
P
(
x
)
are same.
algebra
polynomial