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Baltic Way
1991 Baltic Way
8
A polynomial with four distinct roots
A polynomial with four distinct roots
Source:
April 19, 2013
algebra
polynomial
calculus
derivative
Problem Statement
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
be distinct real numbers. Prove that the equation
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
e
)
(x - a)(x - b)(x - c)(x - d) + (x - a)(x - b)(x - c)(x - e)
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
c
)
(
x
−
e
)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
d
)
(
x
−
e
)
+
(
x
−
a
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
+(x - a)(x - b)(x - d)(x - e) + (x - a)(x - c)(x - d)(x - e)
+
(
x
−
a
)
(
x
−
b
)
(
x
−
d
)
(
x
−
e
)
+
(
x
−
a
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
+
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
=
0
+(x - b)(x - c)(x - d)(x - e) = 0
+
(
x
−
b
)
(
x
−
c
)
(
x
−
d
)
(
x
−
e
)
=
0
has four distinct real solutions.
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