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1993 Poland - First Round
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5
Part of
1993 Poland - First Round
Problems
(1)
Polynomials with 3 real roots
Source: Poland Math Olympiad 1993 Round 1 #5
6/3/2023
Prove that if the polynomial
x
3
+
a
x
2
+
b
x
+
c
x^3 + ax^2 + bx + c
x
3
+
a
x
2
+
b
x
+
c
has three distinct real roots, so does the polynomial
x
3
+
a
x
2
+
1
4
(
a
2
+
b
)
x
+
1
8
(
a
b
−
c
)
x^3 + ax^2 + \frac{1}{4}(a^2 + b)x + \frac{1}{8}(ab-c)
x
3
+
a
x
2
+
4
1
(
a
2
+
b
)
x
+
8
1
(
ab
−
c
)
.
algebra
polynomial