Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases

Question 2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the
following cases:
(i) p(x) = 2x³ + x² – 2x – 1, g(x) = x + 1
(ii) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2
(iii) p(x) = x³ – 4x² + x + 6, g(x) = x – 3

Use the Factor Theorem to determine whether g(x) = x + 1  is a factor of p(x) = 2x³ + x² – 2x – 1
Solution: (i) p(x) = 2x³ + x² – 2x – 1, g(x) = x + 1
Apply remainder theorem
=>x + 1 =0
=> x = - 1
Replace x by – 1 we get
=>2x³ + x² – 2x – 1
=>2(-1)³ + (-1)² -2(-1) - 1
=> -2 + 1 + 2  - 1
=> 0
Remainder is 0 so that x+1 is a factor of 2x³ + x² – 2x – 1

Use the Factor Theorem to determine whether g(x) = x + 2is a factor of p(x) = x³ + 3x² + 3x + 1

(ii) p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2
Apply remainder theorem
=>x + 2 =0
=> x = - 2
Replace x by – 2 we get
=>x³ + 3x² + 3x + 1
=>(-2)³ + 3(-2)² + 3(-2) + 1
=> -8 + 12 - 6 + 1
=> -1
Remainder is not equal to 0 so that x+2 is not a factor of x³ + 3x² + 3x + 1

Use the Factor Theorem to determine whether   g(x) = x – 3is a factor of p(x) = x³ – 4x² + x + 6

 (iii) p(x) = x³ – 4x² + x + 6, g(x) = x – 3
Apply remainder theorem
=>x - 3 =0
=> x = 3
Replace x by – 2 we get
=>x³ – 4x² + x + 6
=>(3)³ -4(3)² + 3 + 6
=> 27  - 36   +3 + 6
=> 0
Remainder is equal to 0 so that x-3 is a factor of  x³ – 4x² + x + 6