by Mathmaven53 » Wed Jan 06, 2016 12:43 pm
Let W(x) be the polynomial. Assume it has integer coefficients.
Divide W(x) by x^2 - 2
W(x) = (x^2 - 2)Q(x) + a x + b
where Q(x) is the quotient polynomial and has integer coefficients and a and b are integers
since x - sqrt 2 divides W(x) then W(sqrt 2) = 0
So W(sqrt 2) = a sqrt 2 + b = 0
Now if a not equal to zero
sqrt 2 = -b/a
which would mean that sqrt 2 is a rational number.
This is false.
So the assumption that a not equal to zero leads to a contradiction
So a = 0
Then from a sqrt 2 + b = 0
b = 0
W(x) = (x^2 - 2) Q(x)
Thus W(x) divisible by x^2 - 2