Given-and-are-the-roots-of-x-3-px-2-qx-pq-0-Find-the-value-of-

Question Number 116221 by bobhans last updated on 02/Oct/20

Given α, β and φ are the roots of

x^{3} - {px}^{2} + qx - pq = 0 .

Find the value of \frac{α}{β} + \frac{β}{α} + \frac{β}{φ} + \frac{φ}{β} + \frac{α}{φ} + \frac{φ}{α} = ?

Answered by TANMAY PANACEA last updated on 02/Oct/20

(\frac{β}{α} + \frac{φ}{α} + 1) + (\frac{α}{β} + \frac{φ}{β} + 1) + (\frac{α}{φ} + \frac{β}{φ} + 1) - 3

= (α + β + φ) (\frac{1}{α} + \frac{1}{β} + \frac{1}{φ}) - 3

= (α + β + φ) (\frac{α β + β φ + α φ}{α β φ}) - 3

= p (\frac{q}{p q}) - 3 = - 2

Answered by ruwedkabeh last updated on 02/Oct/20

\frac{α}{β} + \frac{β}{α} + \frac{β}{φ} + \frac{φ}{β} + \frac{α}{φ} + \frac{φ}{α}

= \frac{α^{2} φ + β^{2} φ + α β^{2} + α φ^{2} + α^{2} β + β φ^{2}}{α β φ}

= \frac{(α + β + φ) (α β + α φ + β φ) - 3 α β φ}{α β φ}

= \frac{(p) (q) - 3 p q}{p q}

= - 2

Answered by bobhans last updated on 02/Oct/20

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