$\sqrt{2}$ cannot represent a rational number I am not asking for a proof that shows me that $\sqrt{2}$ cannot represent a rational number, because I have already seen one by contradiction, which was quite simple, but I have problems in understanding the following proof:


  By Rational Zeros Theorem, the only rational numbers that could possibly be solutions of $x^2 - 2 = 0$ are $\pm1, \pm2$. [Here n = 2, $a_2 = 1$, $a_1 = 1$, $a_0 = -2$. So rational solutions must have the form of $\frac{p}{q}$, where $p$ divides $a_0 = -2$, and $q$ divides $a_2 = 1$.] One can substitute each of the four numbers $\pm1, \pm2$ into the equation $x^2 - 2 = 0$ to quickly eliminate them as possible solutions of the equation. Since $\sqrt{2}$ represents a solutions for $x^2 - 2 = 0$, it cannot represent a rational number.


I have a few questions:


Where does this equation $x^2 - 2 = 0$ come from?
I have read the Rational Zeros Theorem, but I don't understand completely why $\pm1, \pm2$ are the only solutions. 
We want to show that $\sqrt{2}$ cannot represent a rational number, that the proof finishes by saying "Since $\sqrt{2}$ represents a solutions for $x^2 - 2 = 0$, it cannot represent a rational number.", which sincerely I am not seeing well the point. Ok, $\sqrt{2}$ is a solution, but that it cannot represent a rational number from this proof, this seems like not connected at all.


I am know I am missing something that you guys understand on the fly, but of course I am not you :D
HINT: $\alpha=\pm\sqrt{2}$ if and only if $\alpha$ is a solution of $x^2-2=0$. The Rational Zeros Theorem states the following:


  If $p/q$ (where $p$ and $q$ are relatively prime integers) is a root of the polynomial $a_0+a_1x+\cdots +a_nx^n$ (where each $a_i$ is an integer), then $p$ must be a divisor of $a_0$ and $q$ must be a divisor of $a_n$


If you apply this result to the polynomial $x^2-2=0$, you get that any rational root of it must be of the form $p/q$ where $p$ is a divisor of $-2$ and $q$ is a divisor of $1$, what are the possibilities for the rationals roots then? can you find a rational root of this polynomial?
This is a consequence of Gauss' Lemma about content of a polynomial with integer coefficients. Here it means, if a monic polynomial with integer coefficient factorizes over the rationals then it factorizes over integers.
So any factorization of $(x^2-2)$ will be a factorization over integers: now see what are the possibilities for the constant term