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Question
Simplify:
(1 + x)(1 - x)(1 - x + x2)(1 + x + x2)
Solution
(1 + x)(1 - x)(1 - x + x2)(1 + x + x2)
= (1 + x)(1 - x)(x2 + 1 - x)(x2 + 1 + x)
= (12 - x2)[(x2 + 1 - x)2 - x2] .....(Using a2 - b2 = (a + b)(a - b))
= (1 - x2)[x4 + 2x2 + 1 - x2]
= (1 - x2)(x4 + x2 + 1)
= 1(x4 + x2 + 1) -x2(x4 + x2 + 1)
= x4 + x2 + 1 - x6 - x4 - x2
= 1 - x6.
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