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Polynomial has all roots real- OIMU 2005 Problem 7
Polynomial has all roots real- OIMU 2005 Problem 7
Source:
September 3, 2010
algebra
polynomial
real analysis
Rolle s theorem
Problem Statement
Prove that for any integers
n
,
p
n,p
n
,
p
,
0
<
n
≤
p
0<n\leq p
0
<
n
≤
p
, all the roots of the polynomial below are real:
P
n
,
p
(
x
)
=
∑
j
=
0
n
(
p
j
)
(
p
n
−
j
)
x
j
P_{n,p}(x)=\sum_{j=0}^n {p\choose j}{p\choose {n-j}}x^j
P
n
,
p
(
x
)
=
j
=
0
∑
n
(
j
p
)
(
n
−
j
p
)
x
j
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