Answer :
Given:
The function is
[tex]f(x)=2x^2+kx-9[/tex]
The remainder when f(x) is divided by x+6 is 57.
To find:
The value of k.
Solution:
According to the remainder theorem, if a polynomial function f(x) is divided by (x-a), then f(a) is the remainder.
The remainder when f(x) is divided by x+6 is 57.
Using remainder theorem, we get f(-6)=57.
Putting x=-6 in the given function.
[tex]f(-6)=2(-6)^2+k(-6)-9[/tex]
[tex]57=2(36)-6k-9[/tex]
[tex]57=72-6k-9[/tex]
[tex]57=63-6k[/tex]
On further simplification, we get
[tex]6k=63-57[/tex]
[tex]6k=6[/tex]
[tex]k=\dfrac{6}{6}[/tex]
[tex]k=1[/tex]
Therefore, the value of k is 1.