Verify that the division of a polynomial $a(x)$ by a polynomial $b(x) \neq$ 0 can be performed by the following algorithm:
(a) If $b(x)$ has a degree larger than $a(x)$, then the quotient is 0 , and the remainder is $a(x)$.
(b) If $b(x)$ has a degree smaller or equal to that of $a(x)$, divide the leading coefficient $a_n$ of $a(x)$ by the leading coefficient $b_m$ of $b(x)$. The result $q_{n-m}=a_n \div b_m$ is the coefficient of the quotient at the power $x^{n-m}$.
(c) Put $a(x):=a(x)-q_{n-m} x^{n-m} b(x)$ and perform recursively steps (a), (b), and (c). [Observe that the new polynomial $a(x)$ has a smaller degree than the original one. Thus, the algorithm stops after $n$ repetitions of steps (a), (b), and (c).]