Answer:
The value of k is 2
Step-by-step explanation:
The polynomial remainder theorem
The polynomial remainder theorem states that the remainder of the division of a polynomial f(x) by (x-r) is equal to f(r).
The function:
[tex]f(x) = 2x^4 + 9x^3 - 17x^2 + kx -12k[/tex]
has a factor of (x+6).
Applying the polynomial remainder theorem for r=-6, substitute x=-6
[tex]f(-6) = 2(-6)^4 + 9(-6)^3 - 17(-6)^2 + k(-6) - 12k[/tex]
Operating:
[tex]f(-6) = 2*1296 + 9*(-216) - 17*36 + k(-6) - 12k[/tex]
[tex]f(-6) = 2592 - 1944 - 612 -6k - 12k[/tex]
[tex]f(-6) = 36 -18k[/tex]
If x+6 is a factor, then the remainder is zero:
36 -18k=0
Subtracting 36:
-18k=-36
k=-36 / (-18)=2
k=2
The value of k is 2
