Question 1040816
Using the remainder theorem, the idea is that plugging in x = 1 will lead to a result of 2. We need to find the value of k that makes this happen.


{{{f(x) = x^4+k*x^3+1}}}
{{{f(1) = (1)^4+k*(1)^3+1}}} Replace every x with 1
{{{f(1) = 1+k*(1)+1}}}
{{{f(1) = 1+k+1}}}
{{{2 = 1+k+1}}} Replace f(1) with 2. Since we want the remainder to be 2.


Let's solve for k


{{{2 = 1+k+1}}} 
{{{2 = k+2}}} 
{{{k+2 = 2}}} 
{{{k+2-2 = 2-2}}} 
{{{k = 0}}} 


So the final answer is {{{k = 0}}}