Polinomial proof

Polinomial proof

Postby Guest » Sun Jul 08, 2012 6:27 pm

Hi,
I need help with this task.
We've got polinomial W(x). Proof that if [tex]x- \sqrt(2) | W(x)[/tex] then [tex]x^2 -2 | W(x)[/tex].
Thanks for help,
John
Guest
 

Re: Polinomial proof

Postby 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

Mathmaven53
 
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