Question

If α, β, γ are are the zeros of the polynomial f(x) = x³ − px² + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]

Answer 1

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).`