polynomial

Let f(x) = x^3+bx+c  where b and c are integers. If f(5+\sqrt 2)=0 determine b+c

\((5+\sqrt 2)^3+(5+\sqrt 2)b+c=0\\ \text{If a+b= a constant then I can let one of them be b=0 maybe}\\ If\;\;b=0\;\;then (5+\sqrt 2)^3+c=0\\ c=-(5+\sqrt 2)^3\\ b+c=-(5+\sqrt 2)^3\approx -264 \)

\(However\\ If\;\;c=0\\ (5+\sqrt2)^3+(5+\sqrt2)b=0\\ (5+\sqrt2)b=(5+\sqrt2)^3\\ b=(5+\sqrt2)^2\\ so\\ b+c=(5+\sqrt2)^2\)

The answers are different so b+c does not equal a constant.

I expect you did not copy the question properly.

LaTex:

(5+\sqrt 2)^3+(5+\sqrt 2)b+c=0\\ 
\text{If a+b= a constant then I can let one of them be b=0 maybe}\\
If\;\;b=0\;\;then
(5+\sqrt 2)^3+c=0\\ 
c=-(5+\sqrt 2)^3\\
b+c=-(5+\sqrt 2)^3\approx -264