Question

If $\alpha ,\beta ,\gamma$ are the zeros of a polynomial $P\left(x\right)=a{x}^3+b{x}^2+cx+d(a\ne 0)$ then
$\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=$..............

• A. $-\frac{b}{a}$
• B. $-\frac{c}{d}$
• C. $-\frac{b}{c}$
• D. $-\frac{c}{a}$

Answer 1

Answer: (B)

$-\frac{c}{d}$

$\alpha ,\beta$ and $\gamma$ are the zeros of a polynomial

$P\left(x\right)=a{x}^3+b{x}^2+cx+d$

$⟹\alpha \cdot \beta \cdot \gamma =-\frac{d}{a}$ ......... $\left(i\right)$

and $\alpha +\beta +\gamma =\frac{c}{a}$ ......... $\left(ii\right)$

Now, $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=\frac{\alpha \beta +\beta \gamma +\gamma \alpha }{\alpha \beta \gamma }=\frac{c/a}{-d/a}=-\frac{c}{d}$