Question

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x ³ + x ² – 2x – 1, g(x) = x + 1
(ii) p(x) = x ³ + 3x ² + 3x + 1, g(x) = x + 2
(iii) p(x) = x ³ – 4x ² + x + 6, g(x) = x – 3

Answer 1

(i) If g(x) = x + 1 is a factor of the given polynomial p(x), then p(−1) must be zero. 

p(x) = 2x³ + x² − 2x − 1 p(−1) = 2(−1)³ + (−1)² − 2(−1) − 1 = 2(−1) + 1 + 2 − 1 = 0 

Hence, g(x) = x + 1 is a factor of the given polynomial.

(ii) If g(x) = x + 2 is a factor of the given polynomial p(x), then p(−2) must be 0. 

p(x) = x³ +3x² + 3x + 1 p(−2) = (−2)³ + 3(−2)² + 3(−2) + 1 = − 8 + 12 − 6 + 1 = −1 As p ≠ 0, (−2) 

Hence, g(x) = x + 2 is not a factor of the given polynomial.

(iii)If g(x) = x − 3 is a factor of the given polynomial p(x), then p(3) must be 0. 

p(x) = x³ − 4 x² + x + 6 p(3) = (3)³ − 4(3)² + 3 + 6 = 27 − 36 + 9 = 0

Hence, g(x) = x − 3 is a factor of the given polynomial.