If α,βare roots of x2+x+1=0, then α2β+β2α=?

So α+β=−1,αβ=1α2β+β2α=α3+β3αβ=(α+β)3−3(α+β)(αβ)αβ=(−1)3−3(−1)(1)1=−1+31=2

If α,βare roots of x2+x+1=0, then α2β+β2α=?

So α+β=−1,αβ=1α2β+β2α=α3+β3αβ=(α+β)3−3(α+β)(αβ)αβ=(−1)3−3(−1)(1)1=−1+31=2

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If $α, β$ are roots of $x^{2} + x + 1 = 0$ , then $\frac{α^{2}}{β} + \frac{β^{2}}{α} = ?$

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So $α + β = - 1, α β = 1$

$\frac{α^{2}}{β} + \frac{β^{2}}{α} = \frac{α^{3} + β^{3}}{α β} = \frac{{(α + β)}^{3} - 3 (α + β) (α β)}{α β}$

$= \frac{{(- 1)}^{3} - 3 (- 1) (1)}{1} = \frac{- 1 + 3}{1}$ $= 2$

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If $α, β$ are roots of $x^{2} + x + 1 = 0$ , then $\frac{α^{2}}{β} + \frac{β^{2}}{α} = ?$