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Question
Simplify:
`(1 - 2x + x^2)/(1 - x^3) xx (1 + x + x^2)/(1 + x)`
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Solution
It is known that,
a2 − b2 = (a + b) (a − b)
a3 − b3 = (a − b)(a2 + ab + b2)
`(1 - 2x + x^2)/(1 - x^3) xx (1 + x + x^2)/(1 + x)`
= `(1 - x - x + x^2)/((1)^3 - (x)^3) xx (1 + x + x^2)/(1 + x)`
= `(1(1 - x) - x(1 - x))/((1 - x){(1)^2 + (1) xx (x) + (x)^2}) xx ((1 + x + x^2))/(1 + x)`
= `((1 - x) (1 - x))/((1 - x) (1 + x + x^2)) xx ((1 + x + x^2))/((1 + x))`
= `(1 - x)/(1 + x)`
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