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?
+128474 If 1 + sqrt 2 is a root so is 1 - sqrt 2
So
( x - (1 + sqrt 2)) ( x - (1 - sqrt 2)) = x^2 - 2x - 1
x + 2
x^2 - 2x - 1 [ x^3 + 0x^2 + bx + c ]
x^3 -2 x^2 - 1x
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2x^2 + (b + 1)x + c
2x^2 - 4x -2
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(b + 1 + 4)x + c + 2
(5 + b) x + (c + 2) = 0
This will = 0 when b = -5 and c= -2
And the interger root is
x + 2 = 0
x = -2
