Question

If α, β, γ are the zeros of the polynomial f(x) = x³ – px² + qx – r, then (1/αβ) + (1/βγ) + (1/γα) =
A. r/p
B. p/r
C. -p/r
D. -r/p

Answer 1

$\boldsymbol{\frac{1}{\alpha \beta }+\frac{1}{\beta \gamma }+\frac{1 }{\gamma \alpha}=\frac{p}{r}}$  . Option B is correct.

Step-by-step explanation:

• Given polynomial which zeroes are α, β and γ

        $\mathbf{F(x)=x^{3}-px^{2}+qx-r}$

• So

         $\mathbf{\alpha +\beta +\gamma =-\frac{(-p)}{1}=p}$       ...1)

         $\mathbf{\alpha\beta +\beta\gamma +\gamma\alpha =\frac{(q)}{1}=q}$     ...2)

         $\mathbf{\alpha\beta\gamma =-\frac{(-r)}{1}=r}$                ...3)

• Now come to question

        $\boldsymbol{\frac{1}{\alpha \beta }+\frac{1}{\beta \gamma }+\frac{1 }{\gamma \alpha}}$

        Here L.C.M of denominator is αβγ

        $\boldsymbol{\frac{\gamma +\alpha +\beta }{\alpha \beta \gamma }}$        ...4)

• Equation 4) can be written as with help of equation 1) and equation 3)

        $\boldsymbol{\frac{\gamma +\alpha +\beta }{\alpha \beta \gamma }=\frac{p}{r}}$    Means option B is correct.