Answer: k=-5
Step-by-step explanation:
The remainder theorem says that for any polynomial p(x) , if we divide it by the binomial x−a , the remainder is equal to the value of p(a) .
Therefore for [tex]p(x)=x^3+(k+8)x+k[/tex] if it is divided by x-2 and x+1 then its remainder must be p(2) and p(-1)
where, [tex]p(2)=2^3+(k+8)2+k=8+2k+16+k=3k+24[/tex]
[tex]p(-1)=(-1)^3+(k+8)(-1)+k=-1-k-8+k=-9[/tex]
According to the question,
p(2)+p(-1)=0
[tex]\Rightarrow3k+24-9=0\\\Rightarrow3k+15=0\\\Rightarrow3k=-15\\\Rightarrow\ k=\frac{-15}{3}\\\Rightarrow\ k=-5[/tex]
