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Polynomial Divisibility
Polynomial Divisibility
Source: 2008 Greek TST,Pr.1
May 25, 2016
algebra
polynomial
Divisibility
Problem Statement
Find all possible values of
a
∈
R
a\in \mathbb{R}
a
∈
R
and
n
∈
N
∗
n\in \mathbb{N^*}
n
∈
N
∗
such that
f
(
x
)
=
(
x
−
1
)
n
+
(
x
−
2
)
2
n
+
1
+
(
1
−
x
2
)
2
n
+
1
+
a
f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a
f
(
x
)
=
(
x
−
1
)
n
+
(
x
−
2
)
2
n
+
1
+
(
1
−
x
2
)
2
n
+
1
+
a
is divisible by
ϕ
(
x
)
=
x
2
−
x
+
1
\phi (x)=x^2-x+1
ϕ
(
x
)
=
x
2
−
x
+
1
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