(i) If (x + 1) is a factor of p(x) = x^{3} + x^{2} + x + 1, then p (−1) must be zero,

otherwise (x + 1) is not a factor of p(x).

p(x) = x^{3} + x^{2} + x + 1

p(−1) = (−1)^{3} + (−1)^{2} + (−1) + 1

= − 1 + 1 − 1 − 1 = 0

(ii) If (x + 1) is a factor of p(x) = x^{4} + x^{3} + x^{2} + x + 1, then p (−1) must be zero, otherwise (x + 1) is not a factor of p(x).

p(x) = x^{4} + x^{3} + x^{2} + x + 1

p(−1) = (−1)^{4} + (−1)^{3} + (−1)^{2} + (−1) + 1

= 1 − 1 + 1 −1 + 1 = 1 As p(− 1) ≠ 0,

Therefore, x + 1 is not a factor of this polynomial.