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Existence of Limit

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$21 . I f α, β a r e t h e r o o t s o f t h e e q u a t i o n x^{2} - p x + q = 0, t h e n f i n d t h e q u a d r a t i c e q u a t i o n t h e r o o t s o f w h i c h a r e (α^{2} - β^{2}) (α^{3} - β^{3}) a n d α^{3} β^{2} + α^{2} β^{3} .$

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Dear student
$Given : α and β are roots of x^{2} - px + q = 0 α + β = \frac{- coeff . of x}{coeff . of x^{2}} = \frac{- (- p)}{1} = p αβ = \frac{constant term}{coeff . of x^{2}} = \frac{q}{1} = q We know that {(α + β)}^{2} = α^{2} + β^{2} + 2 αβ α^{2} + β^{2} = p^{2} - 2 q and {(α - β)}^{2} = α^{2} + β^{2} - 2 αβ = p^{2} - 2 q - 2 q = p^{2} - 4 q So, (α - β) = \pm \sqrt{p^{2} - 4 q} We want to find the quadratic equation whose roots are : (α^{2} - β^{2}) (α^{3} - β^{3}) and α^{3} β^{2} + α^{2} β^{3} let S and P be the sum and product of roots respectively . S = α^{2} - β^{2} + α^{3} - β^{3} + α^{3} β^{2} + α^{2} β^{3} S = (α + β) (α - β) + (α - β) (α^{2} + β^{2} + αβ) + α^{2} β^{2} (α + β) Case I when (α - β) = \sqrt{p^{2} - 4 q} S = p \times \sqrt{p^{2} - 4 q} + \sqrt{p^{2} - 4 q} (p^{2} - 2 q + q) + q^{2} p = p \sqrt{p^{2} - 4 q} + \sqrt{p^{2} - 4 q} (p^{2} - 2 q + q) + q^{2} p = \sqrt{p^{2} - 4 q} (p + p^{2} - q) + {pq}^{2} and P = (α^{2} - β^{2}) (α^{3} - β^{3}) α^{2} β^{2} (α + β) = (α + β) (α - β) (α - β) (α^{2} + β^{2} + αβ) α^{2} β^{2} (α + β) = p (p^{2} - 4 q) (p^{2} - q) q^{2} p = p^{2} q^{2} (p^{2} - 4 q) (p^{2} - q) So, required polynomial is = x^{2} - Sx + P = x^{2} - (\sqrt{p^{2} - 4 q} (p + p^{2} - q) + {pq}^{2}) x + p^{2} q^{2} (p^{2} - 4 q) (p^{2} - q) Similarly try when (α - β) = - \sqrt{p^{2} - 4 q}$
Regards

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