Find k if x+2 is a factor....

Question

Find k if x+2 is a factor of  x^3-5x^2 +kx-4

Answer 1

Given that  (x^3-5x^2 +kx-4)   has factor   (x+2), then there is a value (ax^2 + bx+c) such that:
    (x+2)(ax^2+bx+c) = (x^3-5x^2 +kx-4)

 Expanded, this is:
     ax^3  +  (2a+b)x^2 + (2b+c)x +  2c   =
       x^3  +  (-5)x^2    +  (k)x   + (-4)  
 
    So,

       a=1

       c= -2
       2a+b=-5   or  2(1)+b=-5   so b=-7

       k = 2b+c   or k=2(-7)+(-2)   so k=-16

 

Checking (very important):

    (x+2)((1)x^2+(-7)x-2)

    x^3 -5x^2 -16x -4       Check!