algebra

Using synthetic division and dividing x^3+kx^2-2x+k+4    by  x+k    results in a remainder of

3k+4     if this remainder is set to = 0    you will find   k = -4/3  will answer this question.

Thanks ElectricPavlov Here is something for you also to learn from :)

Value of k for which x+k is a factor of x^3+kx^2-2x+k+4 is

 

 

Just use the factor theorem

If x+k is a factor then

f(-k)=0

 

\((-k)^3+k(-k)^2-2(-k)+k+4=0\\ -k^3+k^3+2k+k+4=0\\ 3k+4=0\\ k=\frac{-4}{3}\\ \)

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Why does this work... It is very simple.

I'll show an example

\(f(x)=x^2+5x+6\\ f(x)=(x+1)(x+6)\\ f(-1)=(-1+1)(-1+6)=0\\ f(-6)=(-6+1)(-6+6)=0\\ SO\\ \text{If x+a is a factor of of a polynomial then f(-a) must =0 } \)