Simplify:(1 + x)(1 - x)(1 - x + x^2)(1 + x + x^2) - Vidyadip

Question

Simplify:$(1 + x)(1 - x)(1 - x + x^2)(1 + x + x^2)$

✓

Answer

$(1 + x)(1 - x)(1 - x + x^2)(1 + x + x^2)$
$= (1 + x)(1 - x)(x^2 + 1 - x)(x^2 + 1 + x)$
$= (1^2 - x^2)[(x^2 + 1 - x)^2 - x^2] \dots.....($Using $a^2- b^2 = (a + b)(a - b))$
$= (1 - x^2)[x^4 + 2x^2 + 1 - x^2]$
$= (1 - x^2)(x^4 + x^2 + 1)$
$= 1(x^4 + x^2 + 1) -x^2(x^4 + x^2 + 1)$
$= x^4+ x^2+ 1 - x^6 - x^4 - x^2$
$= 1 - x^6.$

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