Question 11
Let $r \gt 0$ be a real number. We call a monic polynomial with complex coefficients $r$-good if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients *primordial* if all of its coefficients have absolute value at most 1.
a. Prove that any 1-good polynomial has a primordial multiple.
b. If $r \gt 1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple.
a. Prove that any 1-good polynomial has a primordial multiple.
b. If $r \gt 1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple.