Consider the following statements in respect of the polynomial 1-x-x^{n}+x^{n+1}, where n is a natural number:<br>1. It is divisible by 1-2x+x^{2}.<br>2. It is divisible by 1-x^{n}.<br>Which of the statements given above is/are correct?
- A. 1 only
- B. 2 only
- C. Both 1 and 2 ✓
- D. Neither 1 nor 2
Correct Answer: C. Both 1 and 2
Explanation
Factorize the polynomial: 1-x-x^n+x^{n+1} = (1-x) - x^n(1-x) = (1-x)(1-x^n). Thus, it is divisible by 1-x^n. Further expanding (1-x^n) yields (1-x)(1+x+...+x^{n-1}), making the full expression (1-x)^2(1+x+...+x^{n-1}). Since (1-x)^2 = 1-2x+x^2, it is divisible by both.
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