Question
What is the value of the following expression:
$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x)$
This question was previously asked in
RRB NTPC 2019 CBT 1 Question Paper (8-Mar-2021) (Shift 2)
The correct answer is
$(1 - x^{16})$
Evaluate Algebraic Expression
To find the value of the expression, we can rearrange the terms and apply the difference of squares formula repeatedly. The difference of squares formula is $a^2 - b^2 = (a - b)(a + b)$.
Step-by-Step Simplification
- Rearrange the given expression: $ (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x) = (1 - x)(1 + x)(1 + x^2)(1 + x^4)(1 + x^8) $
- Apply the difference of squares formula to the first two terms $(1 - x)(1 + x)$: $ (1 - x)(1 + x) = 1^2 - x^2 = 1 - x^2 $ The expression becomes: $ (1 - x^2)(1 + x^2)(1 + x^4)(1 + x^8) $
- Apply the difference of squares formula again to $(1 - x^2)(1 + x^2)$: $ (1 - x^2)(1 + x^2) = 1^2 - (x^2)^2 = 1 - x^4 $ The expression becomes: $ (1 - x^4)(1 + x^4)(1 + x^8) $
- Apply the difference of squares formula to $(1 - x^4)(1 + x^4)$: $ (1 - x^4)(1 + x^4) = 1^2 - (x^4)^2 = 1 - x^8 $ The expression becomes: $ (1 - x^8)(1 + x^8) $
- Apply the difference of squares formula one last time to $(1 - x^8)(1 + x^8)$: $ (1 - x^8)(1 + x^8) = 1^2 - (x^8)^2 = 1 - x^{16} $
Final Result
The simplified value of the expression is $(1 - x^{16})$.
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