Show that the Absolute Value Function Is not Rational
If you want to prove that the absolute value on $\mathbb{R}$ is not a rational function, your starting point was good: you write $$\lvert x\rvert \sum a_i x^i=\sum b_i x^i.$$ Then, you observe that every positive real number is a zero of the polynomial $x \sum a_i x^i-\sum b_i x^i$, so this one should be zero (a non-zero polynomial has only finitely many roots). This implies that $ x\sum a_i x^i=\sum b_i x^i,$ for any real number $x\in \mathbb{R}$, which contradicts the above inequality for $x$ negative.