2024 · Q33
Consider the following in respect of the polynomial x⁴ᵏ + x⁴ᵏ⁺² + x⁴ᵏ⁺⁴ + x⁴ᵏ⁺⁶.
- 1The remainder is zero when the polynomial is divided by x² + 1. 2. The remainder is zero when the polynomial is divided by x⁴ + 1.
Which of the statements given above is / are correct ?
Elementary MathematicsAlgebra
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Modulo x² + 1 (where x² = −1), p = 1 − 1 + 1 − 1 = 0, so statement 1 is true. Modulo x⁴ + 1 (where x⁴ = −1), p = (1 + x²)[(−1)^k + (−1)^{k+1}] = 0 for every k, since the two powers of −1 always cancel — so statement 2 is also true. Both statements are correct: option (c). (A published third-party key marks (a) here, but the k-dependence in statement 2 cancels exactly.)