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National and Regional Contests
Saudi Arabia Contests
Saudi Arabia Pre-TST + Training Tests
2014 Saudi Arabia Pre-TST
2.2
P(x)=prod a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}
P(x)=prod a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}
Source: 2014 Saudi Arabia Pre-TST 2.2
September 13, 2020
algebra
polynomial
Problem Statement
Let
a
1
,
a
2
,
a
3
,
a
4
,
a
5
a_1, a_2, a_3, a_4, a_5
a
1
,
a
2
,
a
3
,
a
4
,
a
5
be nonzero real numbers. Prove that the polynomial
P
(
x
)
=
∏
k
=
0
4
a
k
+
1
x
4
+
a
k
+
2
x
3
+
a
k
+
3
x
2
+
a
k
+
4
x
+
a
k
+
5
P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}
P
(
x
)
=
k
=
0
∏
4
a
k
+
1
x
4
+
a
k
+
2
x
3
+
a
k
+
3
x
2
+
a
k
+
4
x
+
a
k
+
5
, where
a
5
+
i
=
a
i
a_{5+i} = a_i
a
5
+
i
=
a
i
for
i
=
1
,
2
,
3
,
4
i = 1,2, 3,4
i
=
1
,
2
,
3
,
4
, has a root with negative real part.
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