SOLUTION: The polynomial equation {{{x^3 + bx + c = 0}}},where b and c are rational numbers, has {{{5-sqrt( 2 )}}} as a root. It also has an integer root. What is it?

Algebra ->  Radicals -> SOLUTION: The polynomial equation {{{x^3 + bx + c = 0}}},where b and c are rational numbers, has {{{5-sqrt( 2 )}}} as a root. It also has an integer root. What is it?      Log On


   

Question 1156969: The polynomial equation x%5E3+%2B+bx+%2B+c+=+0,where b and c are rational numbers, has 5-sqrt%28+2+%29 as a root. It also has an integer root. What is it?
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


In a polynomial ax%5E3%2Bbx%5E2%2Bcx%2Bd, the sum of the roots is -b/a.

In the polynomial in this problem, there is no x^2 term (i.e., it is 0x^2), so the sum of the roots is 0.

Given that the polynomial is rational and one root is 5-sqrt%282%29, another root is 5%2Bsqrt%282%29.

The sum of those two roots is 10, so the third root is -10.

ANSWER: -10