find the equation whose rootsare multiplied by 3 of those of 2x³+3x-1=0
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Suppose the roots of the original cubic are
α
,
β
and
γ
.
Then:
x
3
+
a
x
2
+
b
x
+
c
=
(
x
−
α
)
(
x
−
β
)
(
x
−
γ
)
x
3
+
a
x
2
+
b
x
+
c
=
x
3
−
(
α
+
β
+
γ
)
x
2
+
(
α
β
+
β
γ
+
γ
α
)
x
−
α
β
γ
So we have:
⎧
⎪
⎨
⎪
⎩
α
+
β
+
γ
=
−
a
α
β
+
β
γ
+
γ
α
=
b
α
β
γ
=
−
c
The cubic we are looking for is:
(
x
−
α
3
)
(
x
−
β
3
)
(
x
−
γ
3
)
=
x
3
−
(
α
3
+
β
3
+
γ
3
)
x
2
+
(
α
3
β
3
+
β
3
γ
3
+
γ
3
α
3
)
x
−
α
3
β
3
γ
3
So the problem essentially boils down to expressing
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