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.

The equation is valid for all $x$ take $x=0$ to get $c=53$. The equation now rewrites as follows $ax^2+bx=3x^2+4x$ and again this is true for all $x$ especially for say $x=-{4\over 3}$ and the equation now looks like ${16a\over 9}-{4b\over 3}=0$ and for $x=1$ we get $a+b=7$ the solution of the system in $(a,b)$ is $(3,4)$
In general if two polynomial are such as $a_nx^n+\cdots+a_0=b_nx^n+\cdots+b_0$ for all x this means $(a_n-b_n)x^n+\cdots+(a_0-b_0)$ has an infinity of solutions while it has degree $n$ and therefore all its coefficients are $0$ which translates into $a_i=b_i\,\forall i$
