To find the value of the expression (1+x)(1+x2)(1+x4)(1+x8)(1−x),

FPSC 5 Years Past Papers Subject Wise (Solved With Details) / All — Multiple-choice practice

  1. A. x ^16 −1
  2. B. 1+x ^16
  3. C. None of these
  4. D. 1−x ^16

Correct answer

1−x ^16

Explanation

Explanation Given: (1+x)(1+x2)(1+x4)(1+x8)(1−x), ( 1 + x ) ( 1 + x 2 ) ( 1 + x 4 ) ( 1 + x 8 ) ( 1 − x ) This is a well-known identity in algebra that builds up as a geometric progression product: Identity: ( 1 − x ) ( 1 + x ) ( 1 + x^ 2 ) ( 1 + x^ 4 ) ( 1 + x^ 8 ) = 1 − x^ 16 ( 1 − x ) ( 1 + x ) ( 1 + x 2 ) ( 1 + x 4 ) ( 1 + x 8 ) = 1 − x 16 So, ( 1 + x ) ( 1 + x^ 2 ) ( 1 + x^ 4 ) ( 1 + x^ 8 ) ( 1 − x ) = 1 − x^ 16 ( 1 + x ) ( 1 + x 2 ) ( 1 + x 4 ) ( 1 + x 8 ) ( 1 − x ) = 1 − x 16

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