Question

Please answer question no.2

Answer 1

Question :

If alpha , beta and gamma are Zeros of polynomial P( x ) = 6x³ + 3x² = 5x + 1, then find the value of alpha^-1 + beta^-1 and gamma^-1.

AnsWer :

- 5.

Solution :

We have, Cubic Polynomial.

$\tt \dagger \: \: \: \: \: 6 {x}^{3} + 3 {x}^{2} = 5x + 1.$

$\tt \dagger \: \: \: \: \: 6 {x}^{3} + 3 {x}^{2} - 5x - 1 = 0.$

General Equation,

$\tt \dagger \: \: \: \: \: {a}^{3} + b {x}^{2} + cx + d$

$\tt \dagger \: \: \: \: \: \red{a \neq0.}$

Where as,

• a = 6.
• b = 3.
• c = - 5.
• d = - 1.

A/Q,

$\tt \dagger \: \: \: \: \: { \alpha }^{ - 1} + { \beta }^{ - 1} + { \gamma }^{ - 1}$

$\tt : \implies \dfrac{1}{ \alpha } + \dfrac{1}{ \beta } + \dfrac{1}{ \gamma }$

$\tt : \implies \dfrac{ \alpha \beta + \beta \gamma + \gamma \alpha }{ \alpha \beta \gamma }$

$\tt : \implies \dfrac{ \dfrac{c}{a} }{ \dfrac{ - d}{a} }$

$\tt : \implies \dfrac{c}{ - d}$

$\tt : \implies \dfrac{ - 5}{1}$

$\tt : \implies - 5.$

Therefore, the value of required answer is - 5.

Some Information :

$\tt \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}$

$\tt \alpha \beta \gamma = \dfrac{-d}{a}$

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