If ax^2+bx+c=0 has two roots alpha,beta then alpha+beta=-b/a , alpha beta=c/a
Now answer the following question
Find the value of alpha cube+beta cube
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Answer:
(\alpha+\beta)(\alpha^2 - \alpha.\beta + \beta^2)(α+β)(α
2
−α.β+β
2
)
Explanation:
Since, we know that,
(\alpha+\beta)^3=\alpha^3+\beta^3+3.\alpha.\beta(\alpha+\beta)(α+β)
3
=α
3
+β
3
+3.α.β(α+β)
\implies \alpha^3+\beta^3 = (\alpha+\beta)^3 - 3.\alpha.\beta(\alpha+\beta)⟹α
3
+β
3
=(α+β)
3
−3.α.β(α+β)
\implies \alpha^3+\beta^3 = (\alpha + \beta)[(\alpha+\beta)^2-3\alpha.\beta]⟹α
3
+β
3
=(α+β)[(α+β)
2
−3α.β]
\implies \alpha^3+\beta^3 = (\alpha + \beta)(\alpha^2+\beta^2+2.\alpha.\beta-3.\alpha.\beta)⟹α
3
+β
3
=(α+β)(α
2
+β
2
+2.α.β−3.α.β)
( Because, (a+b)² = a² + 2ab + b² ),
\implies \alpha^3+\beta^3 = (\alpha + \beta)(\alpha^2+\beta^2-\alpha.\beta)⟹α
3
+β
3
=(α+β)(α
2
+β
2
−α.β)
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