Why are the coefficients always equal? Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$.
Why is it always true that $a = 3, b = 4$ and $c = 53$?
I've seen many examples like this where the coefficients are equated, and was just wondering why that is always true.

Suppose that $ax^{2} + bx + c = 3x^{2} + 4x + 53$ for all $x$, or $ax^{2} + bx + c - (3x^{2} + 4x + 53) = 0$ for all $x$. 
$ax^{2} + bx + c - (3x^{2} + 4x + 53) = 0$ is a polynomial with at most degree $2$ so by the fundamental theorem of algebra it has at most $2$ roots if it is not the zero polynomial. 
But $ax^{2} + bx + c - (3x^{2} + 4x + 53) = 0$ has infintely many roots (it is zero for every value of $x$), so it must be the zero polynomial. Then we have $a=3$, $b=4$ and $c=53$. 
$$ax^{2} + bx + c - (3x^{2} + 4x + 53) = (a-3)x^2+(b-4)x+(c-53)$$
Every coefficient must be equal to zero for it to be the zero polynomial, thus $a-3=0$, and $a=3$. Similiarly, one has $b=4$ and $c=53$. 
