Remainder and Factor Theorems

Question

Given that kx^3+2x^2+2x+3 and kx^3-2x+9 have a common factor, what are the possible values of k?

 

 

Ans: -7 and 5/9

Answer 1

Let f(x) = kx³ + 2x² + 2x + 3     g(x) = kx³ - 2x + 9

 

Let's say that x - c is a common factor of f(x) and g(x).

 

 Then, by the Factor Theorem,  f(c)= 0 and g(c) = 0  So, f(c) = g(c)

 

 kc³ + 2c² + 2c + 3 = kc³ - 2c + 9

 

         2c² + 4c - 6 = 0

 

          c² + 2c - 3 = 0

 

          (c+3)(c-1) = 0      c = -3 or c = 1

 

So, f(-3) = 0 and f(1) = 0

 

     f(-3) = -27k + 18 - 6 + 3 = 0     So, -27k = -15

 

                                                            k = 15/27 = 5/9

     

     f(1) = k + 2 + 2 + 3 = 0    So, k = -7

 

Note:  g(-3) = g(1) = 0  also yields k = 5/9 or -7