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

1.	Determine which of the following polynomials has (x + 1) a factor:(i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1
Answer» (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.

Determine which of the following polynomials has (x + 1) a factor:(i) x3 + x2 + x + 1 (ii) x4 + x3 + x2 + x + 1

Answer»

(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.