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Pan African
2023 Pan-African
5
Polynomial has real and positive roots iff a=b
Polynomial has real and positive roots iff a=b
Source: PAMO 2023 P5
May 22, 2023
algebra
Problem Statement
Let
a
,
b
a, b
a
,
b
be reals with
a
≠
0
a \neq 0
a
=
0
and let
P
(
x
)
=
a
x
4
−
4
a
x
3
+
(
5
a
+
b
)
x
2
−
4
b
x
+
b
.
P(x)=ax^4-4ax^3+(5a+b)x^2-4bx+b.
P
(
x
)
=
a
x
4
−
4
a
x
3
+
(
5
a
+
b
)
x
2
−
4
b
x
+
b
.
Show that all roots of
P
(
x
)
P(x)
P
(
x
)
are real and positive if and only if
a
=
b
a=b
a
=
b
.
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