If α,β are roots of x²+ax+b=0, find value of α⁴+β⁴ in terms of a,b.
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Answer:
Given α,β are roots of the equation x
2
+ax+b=0,
Then, α+β=−a,αβ=b..1).
Now, we to prove
β
α
to be the root of the equation bx
2
+(2b−a
2
)x+b=0, we have to prove that
b(
β
α
)
2
+(2b−a
2
)
β
α
+b=0
or b(α
2
+β
2
)+(2b−a
2
)αβ=0
Now,
b(α
2
+β
2
)+(2b−a
2
)αβ
=b{(α+β)
2
−2α.β}+(2b−a
2
)αβ
=b{(a)
2
−2b}+(2b−a
2
)b [ Using (1)]
=0.
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