= Solution
{c}
Extend the coefficient lists of each <polynomial> by zeros. Then $\rho(p,q)=\sum_{k\geq0}|a_k-b_k|$ is finite because only finitely many terms are nonzero. It is nonnegative and symmetric. Moreover, $\rho(p,q)=0$ holds exactly when every coefficient agrees, which by uniqueness of <polynomial coefficients> is equivalent to $p=q$.
If $r(t)=\sum c_kt^k$, the <triangle inequality> for the <absolute value> gives
$$
\rho(p,r)=\sum_k|a_k-c_k|\leq\sum_k\bigl(|a_k-b_k|+|b_k-c_k|\bigr)=\rho(p,q)+\rho(q,r).
$$
Thus $\rho$ satisfies all the <metric> axioms. It is the <metric> induced by the <polynomial coefficient sum norm>.
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