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CentroAmerican
2016 CentroAmerican
3
3
Part of
2016 CentroAmerican
Problems
(1)
Cyclic permutation of roots
Source: Centroamerican Olympiad 2016, problem 3
6/19/2016
The polynomial
Q
(
x
)
=
x
3
ā
21
x
+
35
Q(x)=x^3-21x+35
Q
(
x
)
=
x
3
ā
21
x
+
35
has three different real roots. Find real numbers
a
a
a
and
b
b
b
such that the polynomial
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
cyclically permutes the roots of
Q
Q
Q
, that is, if
r
r
r
,
s
s
s
and
t
t
t
are the roots of
Q
Q
Q
(in some order) then
P
(
r
)
=
s
P(r)=s
P
(
r
)
=
s
,
P
(
s
)
=
t
P(s)=t
P
(
s
)
=
t
and
P
(
t
)
=
r
P(t)=r
P
(
t
)
=
r
.
algebra
polynomial
roots