Question

x + 1 is a factor of the polynomial
(a) x³ + x² – x + 1
(b) x³ + x² + x + 1
(c) x⁴ + x³ + x² + 1
(d) x⁴ + 3x³ + 3x² + x + 1

Answer 1

The polynomial f(x) which becomes zero when x = −1 is the required polynomial whose factor is x + 1.

(a) Let f(x) = x³ + x² – x + 1

f(−1) = (−1)³ + (−1)² − (−1) + 1
= −1 + 1 + 1 + 1 = 2 ≠ 0

Therefore, x + 1 is not a factor of the polynomial,

(b) Let f(x) = x³ + x² + x + 1

f(−1) = (−1)³ + (−1)² + (−1) + 1
= −1 + 1 − 1 + 1 = 0

Therefore, x + 1 is a factor of the polynomial,

(c) Let f(x) = x⁴ + x³ + x² + 1

f(−1) = (−1)⁴ + (−1)³ + (−1)² + 1
= 1 − 1 + 1 + 1 = 2 ≠ 0

Therefore, x + 1 is not a factor of the polynomial,

(d) Let f(x) = x⁴ + 3x³ + 3x² + x + 1

f(−1) = (−1)⁴ + 3(−1)³ + 3(−1)² + (−1) + 1
= 1 − 3 + 3 − 1 + 1 = 1 ≠ 0

Therefore, x + 1 is not a factor of the polynomial,

Hence, the correct option is (b).