Consider the following in respect of the polynomial x^{4k}+x^{4k+2}+x^{4k+4}+x^{4k+6} :<br>1. The remainder is zero when the polynomial is divided by x^{2}+1.<br>2. The remainder is zero when the polynomial is divided by x^{4}+1.<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
Factoring the polynomial: x^{4k}(1 + x^2 + x^4 + x^6) = x^{4k}(1+x^2)(1+x^4). Since (1+x^2) and (1+x^4) are both factors, the polynomial is exactly divisible by both x^2+1 and x^4+1.
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