Scholarship review question

Quote:

Originally Posted by Solaris123

Hello, Im trying to get the japanese govenment scholarship, I come from Panama and have basicly taugh myself english. Im having problem understanding the next problem or what they are asking, I believe that they say that problem is already integrated and that number are rational but solution to x is not rational thus making the problem impossible. I believe that it might be because I am not understanding what they mean (already had to look up various other parts of other examns because of how they word it) but this one keeps going circularly. can someone explain what they mean and how do I recognized it if they say it differently? sorry for my bad grammar.

Consider the integral expression in x

P=x^3+x^2+ax+1

where a is a rational number.

At a=( needed answer) the value of P is a rational number for any x which satisfies the equation x^2+2x-2= 0, and in this case the value of P is (second answer).

The question is asking you to find P when x is a solution of the equation $x^2+2x-2= 0$ . So the first step is to solve that quadratic equation. Check that the solutions are $x = \pm\sqrt3-1$ . Now take one of those solutions, say $x = \sqrt3-1$ , and substitute it into the equation $P=x^3+x^2+ax+1$ . You get $P = (\sqrt3-1)^3 + (\sqrt3-1)^2 + a(\sqrt3-1)+1$ . Multiply out all those terms and simplify the result. You are told that P is rational for that value of x. So the coefficient of $\sqrt3$ must be 0. That will give you an equation for $a$ ("first answer"). Plug in that value of $a$ into the equation for P to get the "second answer".