if alpha and beta are roots of a^2+bx+c=0 then alpha^3+beta^3=
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Step-by-step explanation:
⇒ α and β are roots of the equation ax
2
+bx+c=0
⇒ αβ=
a
c
------ ( 1 )
⇒ α
3
β
3
=
a
3
c
3
----- ( 2 )
⇒ α+β=
a
−b
------ ( 3 )
⇒ (α+β)
3
=α
3
+β
3
+3αβ(α+β)
⇒ (
a
−b
)
3
=α
3
+β
3
+3(
a
c
)(
a
−b
) [ By using ( 1 ) and ( 3 ) ]
⇒
a
3
−b
3
=α
3
+β
3
−
a
2
3bc
∴ α
3
+β
3
=
a
3
−b
3
+
a
2
3bc
∴ α
3
+β
3
=
a
3
−b
3
+3abc
∴ α
3
+β
3
=
a
3
3abc−b
3
----- ( 4 )
Now,
⇒
α
3
1
+
β
3
1
=
α
3
β
3
α
3
+β
3
=
a
3
c
3
a
3
3abc−b
3
[ By using ( 2 ) and ( 4 ) ]
=
a
3
3abc−b
3
×
c
3
a
3
=
c
3
3abc−b
3
∴
α
3
1
+
β
3
1
==
c
3
3abc−b
3
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