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1970 IMO Longlists
47
The ugly polynomial divisibility
The ugly polynomial divisibility
Source: IMO LongList 1970 - P47
May 22, 2011
algebra
polynomial
calculus
derivative
algebra unsolved
Problem Statement
Given a polynomial
P
(
x
)
=
a
b
(
a
−
c
)
x
3
+
(
a
3
−
a
2
c
+
2
a
b
2
−
b
2
c
+
a
b
c
)
x
2
+
(
2
a
2
b
+
b
2
c
+
a
2
c
+
b
3
−
a
b
c
)
x
+
a
b
(
b
+
c
)
,
P(x) = ab(a - c)x^3 + (a^3 - a^2c + 2ab^2 - b^2c + abc)x^2 +(2a^2b + b^2c + a^2c + b^3 - abc)x + ab(b + c),
P
(
x
)
=
ab
(
a
−
c
)
x
3
+
(
a
3
−
a
2
c
+
2
a
b
2
−
b
2
c
+
ab
c
)
x
2
+
(
2
a
2
b
+
b
2
c
+
a
2
c
+
b
3
−
ab
c
)
x
+
ab
(
b
+
c
)
,
where
a
,
b
,
c
≠
0
a, b, c \neq 0
a
,
b
,
c
=
0
, prove that
P
(
x
)
P(x)
P
(
x
)
is divisible by
Q
(
x
)
=
a
b
x
2
+
(
a
2
+
b
2
)
x
+
a
b
Q(x) = abx^2 + (a^2 + b^2)x + ab
Q
(
x
)
=
ab
x
2
+
(
a
2
+
b
2
)
x
+
ab
and conclude that
P
(
x
0
)
P(x_0)
P
(
x
0
)
is divisible by
(
a
+
b
)
3
(a + b)^3
(
a
+
b
)
3
for
x
0
=
(
a
+
b
+
1
)
n
,
n
∈
N
x_0 = (a + b + 1)^n, n \in \mathbb N
x
0
=
(
a
+
b
+
1
)
n
,
n
∈
N
.
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