Question

If alpha beta gamma are the zeroes of polynomial ax^3+bx^2+cx+d. Find value of 1/alpha+1/beta+1/gamma.

Answer 1

let the zeros be m,n and p instead of alpha beta and gamma
according to question,
m + n + p = -b/a
mn + np + pm = c/a
mnp = -d/a

1/m + 1/n + 1/p
= ( np + mp + mn ) / ( mnp )
= ( c/a ) / ( -d/a )
= - c / d
answer is - c / d

Answer 2

$\bf \red{\frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma } = - \frac{c}{d} }$

Given:

• If $\alpha, \beta , \: and \: \gamma$ are zeros of cubic polynomial $a {x}^{3} + b {x}^{2} + cx + d$.

To find:

• Find the value of $\frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma }$.

Solution:

Concept to be used:

If $\alpha, \beta, \: and \: \gamma$are the zeros of cubic polynomial $\bf a {x}^{3} + b {x}^{2} + cx + d$,where a≠0;then

1. $\alpha + \beta + \gamma = \frac{ - b}{a} ...eq1$
2. $\alpha \beta + \beta \gamma + \alpha \gamma = \frac{c}{a} ...eq2$ and
3. $\alpha \beta \gamma = \frac{ - d}{a} ...eq3$

Step 1:

Rewrite the expression to simplify.

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

or

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

Step 2:

Write the relation of zeros and coefficient of cubic polynomial.

Put values from eq2 and eq3.

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

or

$\frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma } = \frac{c}{a} \times \frac{ - a}{d}$

or

$\frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma } = - \frac{c}{d}$

Thus,

Value is $\bf \frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma } = - \frac{c}{d}$

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