Question

0.17 Factorize x^3+1/x^3-2​

Answer 1

Answer:

w = [1 + i√3]/2

Step-by-step explanation:

$x^3 + 1/x^3 - 2$

Expression becomes, (x^6 - 2 x^3 + 1) / x^3. Now numerator is easily factorisable as (x^3 - 1)^2

So final expression is (x^3 - 1)^2 / x^3.

x^3 - 1 = (x - 1)*(x^2 + x + 1)

So (x^3 - 1)^2 / x^3 = [(x - 1)^2 * (x^2 + x + 1)^2] / x^3

This cannot be factorized further with real coefficients. However if complex coefficients are allowed, we include imaginary cube roots of unity.

i.e.,

x^2 + x + 1 = (x + w)(x + w^2), where

w = [1 + i√3]/2

Hope it helped !