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Austrian-Polish
1981 Austrian-Polish Competition
5
5
Part of
1981 Austrian-Polish Competition
Problems
(1)
P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 with a_i rational has 1 real root
Source: Austrian Polish 1981 APMC
4/26/2020
Let
P
(
x
)
=
x
4
+
a
1
x
3
+
a
2
x
2
+
a
3
x
+
a
4
P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4
P
(
x
)
=
x
4
+
a
1
x
3
+
a
2
x
2
+
a
3
x
+
a
4
be a polynomial with rational coefficients. Show that if
P
(
x
)
P(x)
P
(
x
)
has exactly one real root
ξ
\xi
ξ
, then
ξ
\xi
ξ
is a rational number.
polynomial
rational
algebra