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.