TITLE: Why are the coefficients always equal?
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: The set $\{1,x,x^2,...\}$ is a linear independent set in the vector space of real valued polynomials ($+$ and $\cdot$ defined in the obvious way). Then, by definition of linear independence,
$$
\sum_{i=0}^n a_ix^i = \sum_{j=0}^m b_jx^j \iff a_i = b_j \quad \forall i,j<n=m.
$$