Determine which of the following polynomials has (x + 1) a factor:
(i) 
(ii) 
(iii) 
(iv) 
(i) If (x + 1) is a factor of p (x) = x3 + x2 + x + 1, p (-1) must be zero
Here, p (x) = x3 + x2 + x + 1
p (-1) = (-1)3 + (-1)2 + (-1) + 1
= -1 + 1 – 1 + 1 = 0
Therefore, x + 1 is a factor of this polynomial
(ii) If (x + 1) is a factor of p (x) = x4 + x3 + x2 + x + 1, p (-1) must be zero
Here, p (x) = x4+ x3 + x2 + x + 1
p (-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1
= 1 - 1 + 1 – 1 + 1 ≠ 0
Therefore, x + 1 is not a factor of this polynomial
(iii) If (x + 1) is a factor of p (x) = x4 + 3x3 + 3x2 + x + 1, p (-1) must be zero
Here, p (x) = x4+ 3x3 + 3x2 + x + 1
p (-1) = (-1)4 + 3(-1)3 + 3(-1)2 + (-1) + 1
= 1 - 3 + 3 – 1 + 1 ≠ 0
Therefore, x + 1 is not a factor of this polynomial
(iv) If (x + 1) is a factor of polynomial
p (x) = x3 – x2 – (2 + 
)x + 
, p(-1) must be zero
p (-1) = (-1)3 – (-1)2 – (2 + 
) (-1) + 
= -1 – 1 + 2 + 
= 2
As, p (-1) ≠ 0
Therefore, x + 1 is not a factor of this polynomial