MathDB
Problems
Contests
National and Regional Contests
India Contests
Postal Coaching
2015 Postal Coaching
3
3
Part of
2015 Postal Coaching
Problems
(1)
Monic Polynomials divisible by powers of (x-1)
Source: India Postals 2015
12/2/2015
Let
n
≥
2
n\ge2
n
≥
2
and let
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
be a polynomial with real coefficients. Prove that if for some positive integer
k
(
<
n
)
k(<n)
k
(
<
n
)
the polynomial
(
x
−
1
)
k
+
1
(x-1)^{k+1}
(
x
−
1
)
k
+
1
divides
p
(
x
)
p(x)
p
(
x
)
then
∑
i
=
0
n
−
1
∣
a
i
∣
≥
1
+
2
k
2
n
\sum_{i=0}^{n-1}|a_i| \ge 1 +\frac{2k^2}{n}
i
=
0
∑
n
−
1
∣
a
i
∣
≥
1
+
n
2
k
2
algebra
polynomial