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

Algebra ->  Expressions-with-variables -> SOLUTION: Let $f(x) = x^3+bx+c$, where $b$ and $c$ are integers. If $f(5+\sqrt 3)=0$, determine $b+c$.      Log On


   

Question 1075766: Let $f(x) = x^3+bx+c$, where $b$ and $c$ are integers.
If $f(5+\sqrt 3)=0$, determine $b+c$.
Answer by ikleyn(53937) About Me  (Show Source):
You can put this solution on YOUR website!
.
If you really came from "Art of problem solving" (as the tutor KMST think), it should be an elementary assignment for you.


You are given that 

%285+%2B+sqrt%283%29%29%5E3+%2B+b%2A%285%2Bsqrt%283%29%29+%2B+c = 0    (if I correctly read your formula).


Expand the parentheses. You will get a long expression in the left side of the equation, in which some terms go 
with the multiplier sqrt%283%29, while the other terms go without this multiplier.

Combine the terms in the left side in two groups. One group contains all the terms with the multiplier sqrt%283%29,
while the other group does not contain such terms.


Then your original equation will deploy in TWO equations for each of the two groups separately.


Thus you will get TWO equations for the unknowns "b" and "c".
You will be able easily solve these equations and find "b" and "c" separately. 

Then calculate "b+c".

For the person who got this assignment, these explanations (these hints) are more than enough.


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