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Moldova Team Selection Test
2001 Moldova Team Selection Test
7
P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X)
P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X)
Source: Moldova TST 2001
August 6, 2023
polynomial
Problem Statement
Let
(
P
n
(
X
)
)
n
∈
N
(P_n(X))_{n\in\mathbb{N}}
(
P
n
(
X
)
)
n
∈
N
be a sequence of polynomials defined as:
P
1
(
X
)
=
X
−
1
,
P
2
(
X
)
=
X
2
−
X
−
1
,
P
n
(
X
)
=
X
P
n
−
1
(
X
)
−
P
n
−
2
(
X
)
,
∀
n
>
2
P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X), \forall n>2
P
1
(
X
)
=
X
−
1
,
P
2
(
X
)
=
X
2
−
X
−
1
,
P
n
(
X
)
=
X
P
n
−
1
(
X
)
−
P
n
−
2
(
X
)
,
∀
n
>
2
. For every nonnegative integer
n
n{}
n
find all roots of the polynomial
P
n
(
X
)
P_n(X)
P
n
(
X
)
.
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