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Determine which of the following polynomials has (x + 1) a factor:

(i) x3 + x2 + x + 1 

(ii) x4 + x3 + x2 + x + 1

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(i) If (x + 1) is a factor of p(x) = x3 + x2 + x + 1, then p (−1) must be zero,

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

p(x) = x3 + x2 + 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) = x4 + x3 + x2 + x + 1, then p (−1) must be zero, otherwise (x + 1) is not a factor of p(x).

p(x) = x4 + x3 + x2 + 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.

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