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1978 IMO Longlists
31
P'(x)=nQ(x) for polynomials P, Q
P'(x)=nQ(x) for polynomials P, Q
Source:
October 29, 2010
algebra
polynomial
Problem Statement
Let the polynomials
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
,
P(x) = x^n + a_{n-1}x^{n-1 }+ \cdots + a_1x + a_0,
P
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
⋯
+
a
1
x
+
a
0
,
Q
(
x
)
=
x
m
+
b
m
−
1
x
m
−
1
+
⋯
+
b
1
x
+
b
0
,
Q(x) = x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0,
Q
(
x
)
=
x
m
+
b
m
−
1
x
m
−
1
+
⋯
+
b
1
x
+
b
0
,
be given satisfying the identity
P
(
x
)
2
=
(
x
2
−
1
)
Q
(
x
)
2
+
1
P(x)^2 = (x^2 - 1)Q(x)^2 + 1
P
(
x
)
2
=
(
x
2
−
1
)
Q
(
x
)
2
+
1
. Prove the identity
P
′
(
x
)
=
n
Q
(
x
)
.
P'(x) = nQ(x).
P
′
(
x
)
=
n
Q
(
x
)
.
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