MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1975 Poland - Second Round
6
6
Part of
1975 Poland - Second Round
Problems
(1)
g(x) = f(x) h(x), polynomials
Source: Polish MO Second Round 1975 p6
9/8/2024
Let
f
(
x
)
f(x)
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
be polynomials with integer coefficients. Prove that if for every integer value
n
n
n
the number
g
(
n
)
g(n)
g
(
n
)
is divisible by the number
f
(
n
)
f(n)
f
(
n
)
, then
g
(
x
)
=
f
(
x
)
ā
h
(
x
)
g(x) = f(x)\cdot h(x)
g
(
x
)
=
f
(
x
)
ā
h
(
x
)
, where
h
(
x
)
h(x)
h
(
x
)
is a polynomial,. Show with an example that the coefficients of the polynomial
h
(
x
)
h(x)
h
(
x
)
do not have to be integer.
algebra
polynomial