Question

Show that the polynomial x^4p + x^{4q + 1} + x^{4r + 2} + x^{4s + 3} is divisible by x³ + x² + x + 1 ,where p, q, r, s ∈ N.

Answer 1

Let f(x) = x^4p + x^{4q + 1} + x^{4r + 2} + x^{4s + 3}

Thus by division theorem x³ + x² + x + 1 is factor of  x^4p + x^{4q + 1} + x^{4r + 2} + x^{4s + 3}.