$3\mathrm{.}6$ In Theorem $3\mathrm{.}21,$ we showed that a language is Turing$-$recognizable iff some enu$-$

merator enumerates it$.$ Why didn't we use the following simpler algorithm for the

forward direction of the proof? As before, $s_1,s_2,$dots is a list of all strings in ${}^{**}.$

$E=$ "Ignore the input.

Repeat the following for $i=1,2,3,$dots.

Run $M$ on $s_i.$

If it accepts, print out $s_i."$

$3\mathrm{.}21$ Let $c_1{x}^n+c_2{x}^{n-1}+$cdots$+c_{n}x+c_{n+1}$ be a polynomial with a root at $x=x_0.$ Let

$c_{\text{m}\text{a}\text{x}}$ be the largest absolute value of a $c_i.$ Show that