Prove that if p(x) is polynomial with integer coefficients and p(√2)=0 then p( -√2 )=0.

Prove that if p(x) is polynomial with integer coefficients and p(√2)=0 then p( -√2 )=0.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

## Comments

Sort by:

TopNewestdavid mcmillan see this

Same thing Sandeep Bhardwaj said – Megh Choksi · 2 years, 1 month ago

Log in to reply

– Brian Charlesworth · 2 years, 1 month ago

But \(p(2) = 0\) doesn't imply that \(p(-2) = 0\). How else should I be reading "\(p(2)\) then \(( -2 ) = 0\)" ?Log in to reply

– Megh Choksi · 2 years, 1 month ago

Sir he meant - \(p(\sqrt{2})\)Log in to reply

– David Mcmillan · 2 years, 1 month ago

Sorry for the typo. I meant \( p(-\sqrt{2})\)Log in to reply

To save some time, I'm just going to give you a link to a proof of the general theorem, (which was surprisingly hard to find). Scroll down to theorem 16 on page 14 for the desired proof. You can adapt this proof for the special case \(a = 0, b = 1\) to simplify it a bit. Good question; I don't think I've ever come across a proof of this theorem before. :) – Brian Charlesworth · 2 years, 1 month ago

Log in to reply

Do you mean that if \(p(2) = 0\) then \(p(-2) = 0\)? If so, then this is not the case. If \(p(x) = x^{2} - x - 2\) then \(p(2) = 0\) but \(p(-2) = 4\). Perhaps you meant something else with the terminology "( -2 ) = 0"?

What can be said is that if a complex number \(z\) is a root of a polynomial then so is its conjugate \(\bar {z}\), i.e., \(p(z) = 0 \Leftrightarrow p(\bar {z}) = 0\). – Brian Charlesworth · 2 years, 1 month ago

Log in to reply

@Calvin Lin @brian charlesworth @Sreejato Bhattacharya Please help. – David Mcmillan · 2 years, 1 month ago

Log in to reply

@Sudeep Salgia @Sandeep Bhardwaj Also try and help – David Mcmillan · 2 years, 1 month ago

Log in to reply

– Sandeep Bhardwaj · 2 years, 1 month ago

This is the basic property of a polynomial equation. In fact, this is true for all rational coefficients. You try to prove it by taking a quadratic equation and then try to generalize that.Log in to reply