Correct Answer - Option 1 : (x - 1)
Calculation:
Option A:
P(x) = x - 1
x - 1 = 0
x = 1
f(x) = - x3 - 22x2 + 143x -120
⇒ f(1) = - 1 (1)3 - 22 (1)2 + 143 (1) - 120
⇒ f(1) = - 1 - 22 + 143 - 120
⇒ f(1) = - 143 + 143 = 0
∴ (x - 1) is a factor of (- x3 - 22x2 + 143x -120)
Option B:
P(x) = x + 1
x + 1 = 0
x = - 1
f(x) = - x3 - 22x2 + 143x -120
⇒ f(-1) = - 1 (-1)3 - 22 (-1)2 + 143 (-1) - 120
⇒ f(-1) = 1 - 22 - 143 - 120
⇒ f(-1) = 1 - 285 = -284
∴ (x + 1) is not a factor of (- x3 - 22x2 + 143x - 120)
Option C:
P(x) = x - 2
x - 2 = 0
x = 2
f(x) = - x3 - 22x2 + 143x -120
⇒ f(2) = -1(2)3 - 22(2)2 + 143(2) - 120
⇒ f(2) = -8 - 88 + 286 - 120
⇒ f(2) = -216 + 286 = 70
∴ (x - 2) is not a factor of (- x3 - 22x2 + 143x -120)
Option D:
P(x) = x - 3
x - 3 = 0
x = 3
f(x) = - x3 - 22x2 + 143x -120
⇒ f(3) = -1(3)3 - 22(3)2 + 143(3) - 120
⇒ f(3) = - 27 - 198 + 429 - 120
⇒ f(3) = -345 + 429
⇒ f(3) = 84
∴ (x - 3) is not a factor of (- x3 - 22x2 + 143x -120)
The correct option is 1 i.e. (x - 1)