Question

For what value of n is the polynomial f(x) = 2x^(3) – nx^(2) + 5x + 9 exactly divisible by x + 2?

Answer 1

   f (x) = 2 x³ - n x²  + 5 x + 9
        divisor  = x + 2

     f (-2)  = reminder when f(x) is divided by  x+2
             = 2 (-2)³ - n (-2)² +  5 (-2)  +  9
             = -16 - 4 n - 10 + 9
             =   - 4n -17
     if reminder is to be 0,  then    n = - 17/4

Answer 2

Remainder when f(x) is divisible by (x+a) is f(-a).

f(-2) = 2(-2)³ - n(-2)² + 5(-2) + 9
f(-2) = 2(-8) - n(4) - 1
f(-2) =  -16 - 4n - 1
f(-2) = -17 - 4n

To divide exactly, the remainder has to be equal to 0

-17 - 4n = 0
-4n = 17

n = -17/4