# If α, β, γ are are the zeros of the polynomial f(x) = x3 − px2 + qx − r,

Question:

If $\alpha, \beta, y$ are are the zeros of the polynomial $f(x)=x^{3}-p x^{2}+q x-r$, then $\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}=$

Solution:

We have to find the value of $\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}$

Given $\alpha, \beta, \gamma$ be the zeros of the polynomial $f(x)=x^{3}-p x^{2}+q x-r$

$\alpha+\beta+\gamma=\frac{-\text { Coefficient of } x^{2}}{\text { Coefficient of } x^{3}}$

$=\frac{-(p)}{1}$

$=p$

$\alpha \beta \gamma=\frac{-\text { Constant term }}{\text { Coefficient of } x^{3}}$

$=\frac{-(r)}{1}$

$=r$

Now we calculate the expression

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

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

$\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}=\frac{p}{r}$

Hence, the correct choice is (b)

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