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National and Regional Contests
PEN Problems
PEN Q Problems
7
7
Part of
PEN Q Problems
Problems
(1)
Q 7
Source:
5/25/2007
Let
f
(
x
)
=
x
n
+
5
x
n
ā
1
+
3
f(x)=x^{n}+5x^{n-1}+3
f
(
x
)
=
x
n
+
5
x
n
ā
1
+
3
, where
n
>
1
n>1
n
>
1
is an integer. Prove that
f
(
x
)
f(x)
f
(
x
)
cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
algebra
polynomial
calculus
integration
number theory
relatively prime
Rational Root Theorem