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If \( \alpha, \beta, \gamma \) are the roots of the quadratic equation \( x^{3}-p x^{2}+q x-r=0 \), then the value of \( \sum \alpha^{2} \beta \) is equal |
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Answer» α β and γ are roots of cubic equation x3 - px2 + qx - r = 0 \(\therefore\) sum of roots = \(\cfrac{-(-p)}1\) = p \(\therefore\) α + β + γ = p....(1) sum of product of two roots = q/1 = q \(\therefore\) αβ + βγ + γα = q....(2) products of roots = \(\cfrac{-(-r)}1\) = r \(\therefore\) αβγ = r....(3) multiplying (1) and (2) we get pq = ( α + β + γ) (αβ + βγ + γα) = pq = α2β + αβγ + α2γ + αβ2+ β2γ + αβγ + αβγ + βγ2 +γ2α = (α2β + β2γ + α2γ + α2γ + γ2β + β2α) + 3αβγ \(\sum\)= α2β + 3r (\(\because\) r = αβγ) \(\sum\) α2β = pq -3r |
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