Question

If alpha, beta, gamma are the zeros of the polynomial p(x)=axcube+bxsqure+cx+d then 1/alpha+1/beta+1/gamma=?

Answer 1

Question -

$If \: \alpha , \: \beta \: and \: \gamma \: are \: the \: zeros \: of \: the \\ polynomial \: p(x) = a {x}^{3} + b {x}^{2} + cx + d \\ then \: \frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma }$

Solution -

$\alpha, \: \: \beta, \: \gamma \: are \: the \: zeros \: of \: p(x) = a {x}^{3} + b{x}^{2} + cx + d$

Then,

$\alpha + \beta + \gamma = \frac{ - b}{a}$

$\alpha \beta \gamma = \frac{ - d}{a} ........1)$

And

$\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} ......2)$

Now,

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

$= \frac{ \alpha \beta + \beta \gamma + \gamma \alpha }{ \alpha \beta \gamma }$ [By 1 and 2]

$= \frac{ \frac{c}{a} }{ \frac{ - d}{a} }$

$= \frac{c}{a} \div \frac{ - d}{a}$

$= \frac{c}{a} \times \frac{ - a}{d}$

$= \frac{ - c}{d}$