Consider the following in respect of the polynomial x4k+ x4k+2+ x4k+4+ x4k+6.
1. The 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 ?
Given:
x4k+ x4k+2+ x4k+4+ x4k+6.
Solution:
We have,
x4k+ x4k+2+ x4k+4+ x4k+6
=> x4k (x0 + x2) + x4k+4 (x0 + x2)
=> ( x4k+ x4k+4) (x0 + x2)
=> x4k (x0 + x4) (x0 + x2)
=> x4k (1+ x4) (1+ x2)
After factorizing the polynomial, we can see the given polynomial is a multiple of (1 + x2) and (1 + x4).
Thus, the given polynomial is divisible by both (1 + x2) and (1 + x4).
Hence, both statements are correct.
∴ The correct answer is option (c).
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