Question

If α, β, γ are the zeros of the polynomial p(x) = 6x³ + 3x² – 5x + 1, find the value of `(1/α + 1/β + 1/γ)`.

Answer 1

Given: α, β, γ are the zeros of p(x) = 6x³ + 3x² – 5x + 1.

Step-wise calculation:

1. By Vieta’s formulas for ax³ + bx² + cx + d.

`α + β + γ = (-b)/a` 

`αβ + βγ + γα = c/a` 

`αβγ = (-d)/a`

2. Here a = 6, b = 3, c = –5, d = 1.

So `αβ + βγ + γα = c/a = (-5)/6`.

`αβγ = (-d)/a = (-1)/6`.

3. The sum of reciprocals is

`(1/α + 1/β + 1/γ) = (αβ + βγ + γα)/(αβγ)` 

= `((-5)/6)/((-1)/6)`

= 5