Question : Simplify the following expression:
$[(1 + p)(1 + p^2)(1 + p^4)(1 + p^8)(1 + p^{16})(1 - p) - 1]$
Option 1: $-p^{32}$
Option 2: $p^{32}$
Option 3: $(1 + p^{32})$
Option 4: $(1 - p^{32})$
Correct Answer: $-p^{32}$
Solution :
Given: $[(1+p)(1+p^2)(1+p^4)(1+p^8)(1+p^{16})(1 - p) - 1]$
= $[(1+p)(1-p)(1+p^2)(1+p^4)(1+p^8)(1+p^{16})-1]$
= $[(1-p^2)(1+p^2)(1+p^4)(1+p^8)(1+p^{16})-1]$
= $[(1-p^4)(1+p^4)(1+p^8)(1+p^{16})-1]$
= $[(1-p^8)(1+p^8)(1+p^{16})-1]$
= $[(1-p^{16})(1+p^{16})-1]$
$ = [(1 - p^{32}) -1]$
$ = -p^{32}$
Hence, the correct answer is $-p^{32}$.
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