Question

Simplify:

\[\frac{1 - 2x + x^2}{1 - x^3} \times \frac{1 + x + x^2}{1 + x}\]

Answer 1

It is known that,

a² − b² = (a + b) (a − b)

a³ − b³ = (a − b)(a² + ab + b²)

\[\  \frac{1 - 2x + x^2}{1 - x^3} \times \frac{1 + x + x^2}{1 + x}\]

\[ = \frac{1 - x - x + x^2}{\left(1 \right)^3 - \left(x \right)^3} \times \frac{1 + x + x^2}{1 + x}\]

\[ = \frac{1\left(1 - x \right) - x\left(1 - x \right)}{\left(1 - x \right)\left\{ \left(1 \right)^2 + \left(1 \right) \times \left( x \right) + \left(x \right)^2 \right\}}   \times   \frac{\left(1 + x + x^2 \right)}{1 + x}\]

\[ = \frac{\left(1 - x \right)\left( 1 - x \right)}{\left(1 - x \right)\left(1 + x + x^2 \right)} \times \frac{\left(1 + x + x^2 \right)}{\left(1 + x \right)}\]

\[ = \frac{1 - x}{1 + x}\]