Symmetric Polynomials .

Algebra Level 4

Let α \alpha and β \beta be the roots of the equation x 2 − ( p − q ) x − p = 0 { x }^{ 2 } - (p - q) x - p = 0 . Find the value of

α 3 + α 2 − α − 1 α 3 + α 2 + α ( q − 2 ) − q + β 3 + 3 β 2 + 3 β + 1 β 3 + 3 β 2 + β ( q + 2 ) + q \large \frac { { \alpha }^{ 3 } + { \alpha }^{ 2 } - \alpha - 1 }{ { \alpha }^{ 3 } + { \alpha }^{ 2 } + \alpha( q - 2 ) - q } + \frac { { \beta }^{ 3 } + 3{ \beta }^{ 2 } + 3\beta + 1 }{ { \beta }^{ 3 } + { 3\beta }^{ 2 } + \beta ( q + 2 ) + q }


The answer is 1.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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2 solutions

Vitor Juiz
Feb 8, 2018

First part First part Second part Second part

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Aaghaz Mahajan
Feb 8, 2018

For a shortcut, assume p and q to be both equal to 4......

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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