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Byju's Answer
Standard X
Mathematics
Relationship Between Zeroes and Coefficients of a Cubic Polynomial
If α, β, γ ...
Question
If
α
,
β
,
γ
are the zeroes of the cubic polynomial
x
3
+
4
x
+
2
, then find the value of:
1
α
+
β
+
1
β
+
γ
+
1
γ
+
α
Open in App
Solution
We have,
P
(
x
)
=
x
3
+
4
x
+
2
Since,
α
,
β
,
γ
are the zeroes of this polynomial.
Now,
α
+
β
+
γ
=
−
0
1
=
0
α
β
+
β
γ
+
γ
α
=
4
1
=
4
α
β
γ
=
−
2
1
=
−
2
Since,
1
α
+
β
+
1
β
+
γ
+
1
γ
+
α
⇒
(
β
+
γ
)
(
γ
+
α
)
+
(
α
+
β
)
(
γ
+
α
)
+
(
α
+
β
)
(
β
+
γ
)
(
α
+
β
)
(
β
+
γ
)
(
γ
+
α
)
⇒
β
γ
+
α
β
+
γ
2
+
α
γ
+
α
γ
+
α
2
+
β
γ
+
α
β
+
α
β
+
α
γ
+
β
2
+
β
γ
(
α
+
β
)
(
β
+
γ
)
(
γ
+
α
)
⇒
α
2
+
β
2
+
γ
2
+
3
(
α
β
+
β
γ
+
γ
α
)
(
α
+
β
)
(
β
+
γ
)
(
γ
+
α
)
⇒
α
2
+
β
2
+
γ
2
+
2
(
α
β
+
β
γ
+
γ
α
)
+
(
α
β
+
β
γ
+
γ
α
)
(
α
+
β
)
(
β
+
γ
)
(
γ
+
α
)
⇒
(
α
+
β
+
γ
)
2
+
(
α
β
+
β
γ
+
γ
α
)
(
α
+
β
)
(
β
+
γ
)
(
γ
+
α
)
⇒
0
+
0
(
α
+
β
)
(
β
+
γ
)
(
γ
+
α
)
⇒
0
Hence, this is the answer.
Suggest Corrections
1
Similar questions
Q.
If the
α
,
β
,
γ
are the zeroes of polynomial
P
(
x
)
=
x
3
+
4
x
+
2
, then find the value of
1
α
+
β
+
1
β
+
γ
+
1
γ
+
α
Q.
If
α
,
β
,
γ
are roots of
x
3
+
4
x
+
1
=
0
, then
(
α
+
β
)
−
1
+
(
β
+
γ
)
−
1
+
(
γ
+
α
)
−
1
equals
Q.
If α, β, γ are the roots of the equation
x
3
+
4
x
+
1
=
0
,then
(
α
+
β
)
−
1
+
(
β
+
γ
)
−
1
+
(
γ
+
α
)
−
1
=
Q.
If
α
,
β
,
γ
are zeroes of cubic polynomial
x
3
+
p
x
2
+
q
x
2
+
2
such that
α
β
+
1
=
0
.
Find the value of
2
p
+
q
+
5
.
Q.
If
α
,
β
,
γ
are the roots of
x
3
−
x
2
−
1
=
0
, then find the value of
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1
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1
−
β
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1
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