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

(i) x3 + x2 + x + 1

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

(iii) x4 + 3x3 + 3x2 + x + 1 

(iv) x3 - x2 - (2 + 2)x + 2

Solution:

If (x + 1) is a factor of p(x), p(-1) must be zero.

(i) 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 x3 + x2 + x + 1.

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

p(-1) = (-1)4 + (-1)3 + (-1)2 + (-1) + 1

= 1 - 1 + 1 - 1 + 1 = 1≠ 0

Therefore, x + 1 is not a factor of x4 + x3 + x2 + x + 1.

(iii) p(x) = x4 + 3x3 + 3x2 + x + 1

p(-1) = (-1)4 + 3(-1)3 + 3(-1)2 + (-1) + 1

= 1 - 3 + 3 - 1 + 1 = 1≠ 0

Therefore, + 1 is not a factor of x4 + 3x3 + 3x2 + x + 1.

(iv) p(x) = x3 - x2 - (2 + 2)x + 2

p(-1) =  (-1)3 -  (-1)2 -  (2 + 2) (-1) + 2

= -1 - 1 + 2 + 2 + 2

=22 ≠ 0

Therefore, x + 1 is not a factor of x3 - x2 - (2 + 2)x + 2.