If 0 and 1 are the zeroes of f(x) =3x^3 -5x^2+px+q , find the value of p and q
Answers
Answered by
1
Denoting the three zeros by
α
,
β
,
γ
, we have:
f
(
x
)
=
x
3
−
3
p
x
2
+
q
x
−
r
f
(
x
)
=
(
x
−
α
)
(
x
−
β
)
(
x
−
γ
)
f
(
x
)
=
x
3
−
(
α
+
β
+
γ
)
x
2
+
(
α
β
+
β
γ
+
γ
α
)
x
−
α
β
γ
In particular, equating the coefficient of
x
2
we have:
α
+
β
+
γ
=
3
p
If
α
,
β
and
γ
are in arithmetic progression with common difference
δ
, then:
α
=
β
−
δ
γ
=
β
+
δ
So:
3
β
=
(
β
−
δ
)
+
β
+
(
β
+
δ
)
=
α
+
β
+
γ
=
3
p
So:
β
=
p
That is:
p
is one of the zeros of
f
(
x
)
So:
0
=
f
(
p
)
=
p
3
−
3
p
p
2
−
q
p
−
r
=
−
2
p
3
−
p
q
−
r
Inverting the signs, that is:
2
p
3
+
p
q
+
r
=
0
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