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Putnam
1946 Putnam
A2
A2
Part of
1946 Putnam
Problems
(1)
Putnam 1946 A2
Source: Putnam 1946
3/10/2022
If
a
(
x
)
,
b
(
x
)
,
c
(
x
)
a(x), b(x), c(x)
a
(
x
)
,
b
(
x
)
,
c
(
x
)
and
d
(
x
)
d(x)
d
(
x
)
are polynomials in
x
x
x
, show that
∫
1
x
a
(
x
)
c
(
x
)
d
x
⋅
∫
1
x
b
(
x
)
d
(
x
)
d
x
−
∫
1
x
a
(
x
)
d
(
x
)
d
x
⋅
∫
1
x
b
(
x
)
c
(
x
)
d
x
\int_{1}^{x} a(x) c(x)\; dx\; \cdot \int_{1}^{x} b(x) d(x) \; dx - \int_{1}^{x} a(x) d(x)\; dx\; \cdot \int_{1}^{x} b(x) c(x)\; dx
∫
1
x
a
(
x
)
c
(
x
)
d
x
⋅
∫
1
x
b
(
x
)
d
(
x
)
d
x
−
∫
1
x
a
(
x
)
d
(
x
)
d
x
⋅
∫
1
x
b
(
x
)
c
(
x
)
d
x
is divisible by
(
x
−
1
)
4
.
(x-1)^4.
(
x
−
1
)
4
.
Putnam
algebra
polynomial