Question

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

(i) x 3 + x 2 + x + 1

(ii) x4 + x 3 + x 2 + x + 1

(iii) x 4 + 3x 3 + 3x 2 + x + 1

(iv) x 3 – x 2 – ( 2 + √2 )x + √2

Answer 1

(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

Hence, x + 1 is a factor of this polynomial.

(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 ≠ 0, (− 1)

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

(iii) If (x + 1) is a factor of polynomial p(x) = x4 + 3x3 + 3x2 + x + 1,

then p(−1) must be 0, otherwise (x + 1) is not a factor of this polynomial.

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

= 1 − 3 + 3 − 1 + 1

= 1 As p ≠ 0, (−1) .

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

(iv) If (x + 1) is a factor of polynomial p(x) = must be 0, otherwise (x + 1) is not a factor of this polynomial.

P(-1) = (-1)3- (-1)2- (2 + √2)(-1) + √2 = -1-1+2+√2+√2 = 2√2

As p ≠ 0, (-1). Therefore, (x+1) is not a factor of this polynomial.