The polynomial equation x^3 + bx + c = 0 where b and c are rational numbers, has 1 + sqrt(2) as a root. It also has an integer root. What is it?
+128448 The roots are 1 + sqrt 2 ,1 - sqrt 2 and r
Sum of the products of the roots taken two at a time = 2r - 1 = b
Product of the roots = -r = -c ⇒ r = c
So
x^3 + (2r - 1) x + r = 0 since 1 + sqrt 2 is a root....then
(1 + sqrt 2)^3 + (2r - 1)(1 + sqrt 2) + r = 0
1 + 3sqrt 2 + 3*2 + 2sqrt 2 + 2r - 1 + 2rsqrt 2 - sqrt 2 + r = 0
6 + 4sqrt 2 + 2r sqrt 2 + 3r = 0
6 + 3r + 2sqrt 2 ( 2 + r) = 0
3 ( 2 + r) + 2sqrt 2 ( 2 + r) = 0
(2 + r) ( 3 + 2sqrt 2) = 0
r = -2
So b = 2(-2) - 1 = - 5
c = -2 = r
The polynomial is
x^3 - 5x - 2 and the integer root = -2
