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Moldova Contests
Moldova Team Selection Test
1996 Moldova Team Selection Test
5
5
Part of
1996 Moldova Team Selection Test
Problems
(1)
Find all polynomials $P(X)$ of fourth degree with real coefficients that verify
Source: Moldova TST 1996
8/8/2023
Find all polynomials
P
(
X
)
P(X)
P
(
X
)
of fourth degree with real coefficients that verify the properties: a)
P
(
−
x
)
=
P
(
x
)
,
∀
x
∈
R
;
P(-x)=P(x), \forall x\in\mathbb{R};
P
(
−
x
)
=
P
(
x
)
,
∀
x
∈
R
;
b)
P
(
x
)
≥
0
,
∀
x
∈
R
;
P(x)\geq0, \forall x\in\mathbb{R};
P
(
x
)
≥
0
,
∀
x
∈
R
;
c)
P
(
0
)
=
1
;
P(0)=1;
P
(
0
)
=
1
;
d)
P
(
X
)
P(X)
P
(
X
)
has exactly two local minimums
x
1
x_1
x
1
and
x
2
x_2
x
2
such that
∣
x
1
−
x
2
∣
=
2.
|x_1-x_2|=2.
∣
x
1
−
x
2
∣
=
2.
algebra
polynomial