Shifting quadratic polynomials

If two polynomials are equal for all $x$ , their coefficients are the same.

To see this, suppose you have some polynomials $p(x)=\sum_{i=0}^\infty a_i x^i$ and $q(x)=\sum_{i=0}^\infty b_i x^i$ . First suppose $p(x)=q(x)$ for all $x$ . This must therefore be true when $x=0$ , i.e. $p(0)=q(0)$ . All polynomials are infinitely differentiable everywhere, so for any $k$ we can differentiate $p(x)-q(x)=0$ and find $\left.\frac{\mathrm{d}^k}{\mathrm{d}x^k} (p(x)-q(x))\right|_{x=0}=k! (a_k-b_k) = 0$ so $a_k = b_k$ for all $k$ . There's probably an easier way to show that!

In this case, if $f(x)=g(x)$ for all $x$ , then we must have $a=b$ , $b=c$ and $c=a$ . This means the coefficients cannot be distinct.