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Putnam Exam Seminar
Assignment 9
Fall 2010
Due November 8
Exercise 1.
Do there exist polynomials
a
(
x
)
,b
(
x
)
,c
(
y
)
,d
(
y
) such that
1 +
xy
+
x
2
y
2
=
a
(
x
)
c
(
y
) +
b
(
x
)
d
(
y
)
holds identically? [Putnam Exam, 2003, B1]
Exercise 2.
Let
n
be a positive integer and deﬁne
f
(
n
) = 1! + 2! + 3! +
···
+
n
!
.
Find polynomials
P
(
x
) and
Q
(
x
) such that
f
(
n
+ 2) =
P
(
n
)
f
(
n
+ 1) +
Q
(
n
)
f
(
n
)
for all
n
≥
1.
Exercise 3.
Let
P
(
x
) =
c
n
x
n
+
c
n

1
x
n

1
+
···
+
c
0
be a polynomial with integer coeﬃcients.
Suppose that
r
is a rational number such that
P
(
r
) = 0. Show that the
n
numbers
c
n
r,c
n
r
2
+
c
n

1
r,c
n
r
3
+
c
n

1
r
2
+
c
n

1
r,.
..,c
n
r
n
+
c
n

1
r
n

1
+
···
c
1
r
are integers. [Putnam Exam, 2004, B1]
Exercise 4.
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This note was uploaded on 02/29/2012 for the course MATH 1190 taught by Professor Ryandaileda during the Fall '10 term at Trinity University.
 Fall '10
 RyanDaileda
 Polynomials

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