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Question
If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]
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Solution
We have to find the value of `1/(alphabeta)+1/(betay)+1/(yalpha)`
Given `alpha,beta,y` be the zeros of the polynomial f(x) = x3 − px2 + qx − r,
`alpha + ß + y = (-text{coefficient of }x^2)/(text{coefficient of } x^3)`
`= (-p)/1`
`= p`
`alphabetay= (-\text{Constant term})/(\text{Coefficient of}x^3)`
`(-(r))/1`
`= r`
Now we calculate the expression
`1/(alphabeta)+1/(betay)+1/(yalpha)= y/(alphabetay)+alpha/(alphabetay)+beta/(alphabetay)`
`1/(alphabeta)+1/(betay)+1/(yalpha)= (alpha+y+beta)/(alphabetay)`
`1/(alphabeta)+1/(betay)+1/(yalpha)= p/r`
Hence, the correct choice is `(b).`
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