33.6. To compute the value of the polynomial $P(x)=a_{n} x^{n}+$ $+a_{n-1} x^{n-1}+\ldots+a_{0}$ at $x=x_{0}$, one can compute $a_{n} x_{0}^{n}$, $a_{n-1} x_{0}^{n-1}, \ldots, a_{1} x_{0}$, and then add all the obtained numbers and $a_{0}$. This requires $2 n-1$ multiplications (computing $x_{0}^{k}$ for $k=2,3, \ldots, n$ requires $n-1$ multiplications). Devise a method for computing $P\left(x_{0}\right)$ that requires only $n$ multiplications and $n$ additions (Horner's scheme).
33.6. Let $b_{1}=a_{n} x_{0}+a_{n-1}, b_{2}=b_{1} x_{0}+a_{n-2}, \ldots, b_{n}=b_{n-1} x_{0}+a_{0}$. Then $b_{n}=P\left(x_{0}\right)$, since

$$
P(x)=\left(\ldots\left(\left(a_{n} x_{0}+a_{n-1}\right) x_{0}+a_{n-2}\right) \ldots\right) x_{0}+a_{0}
$$