Irrational polynomial equation

[ciel]

what's the polynomial equation which sqrt2 + sqrt3 satisfies ?

 

[ZioX]

How about x-sqrt2-sqrt3?

 

[Diffy]

There is not one unique polynomial that (sprt2 + sqrt3) satisfies. Perhaps you should elaborate a bit as to what terms, degree, etc. that you require in your polynomial.

 

[Hurkyl]

And keep in mind that a polynomial is nothing more than a linear combination of monomials...

 

[Gib Z]

What about, $$x- \pi$$ ? or $$x^3 - 31$$ =] ?

 

[AlphaNumeric2]

what's the polynomial equation which sqrt2 + sqrt3 satisfies ?

If you want to get rid of the roots, then it'd be something like

$$x = \sqrt{2}+\sqrt{3}$$
$$x^{2} = 2+3+2\sqrt{6}$$
$$x^{2}-5 = 2\sqrt{6}$$
$$(x^{2}-5)^{2} = 24$$

Rearrange and clean up a bit.

 

[ciel]

yeah, i seems alright that way. well, i has to be polynomial, but i couldn't just go upto that part, so i was confused. thnx anyway :)

 

[EES]

What about, $$x- \pi$$ ? or $$x^3 - 31$$ =] ?

 

[robert Ihnot]

Hurkyl: And keep in mind that a polynomial is nothing more than a linear combination of monomials...

From the standpoint of Galois Theory, we can build that form from the conjugates of the two order two equations, which produces an order 4 equation:

The four products of the form: $$\prod (X-(\pm\sqrt2\pm\sqrt3)$$ = X^4-10X^2+1.

 

[HallsofIvy]

It would have been nice if you had posted the actual question. I suspect it was NOT "Find a polynomial equation that $\sqrt{2}+ \sqrt{3}$ satisfies". I suspect rather that it was something like "Find a polynomial equation, with integer coefficients, that $\sqrt{2}+ \sqrt{3}$ satisfies".

 

[Gib Z]

I'm just joking with you man, try it on yours calculator, $\sqrt{2}+\sqrt{3}$ is a valid approximation of pi to several digits. Similar thing for the other equation.