Simplifying a Boolean expression that has 3 variables in it
Take your answer:
$$ z = AC + BC + \overline B \, \overline C 
$$

Now make the following transformation:
$$ z = AC(B+\overline B) + BC + \overline B \, \overline C 
$$
Expand:
$$ z = ABC+A\overline B C + BC + \overline B \, \overline C 
$$
Use the redundancy rule \$X+XY=X\$ on the first and third terms:

$$ z = BC + A\overline B C + \overline B  \overline C = $$
$$ = BC + \overline B(AC+\overline C) $$

Use another rule: \$X+\overline XY=X + Y\$ on the expression in parentheses:
$$ z = BC + \overline B(A+\overline C) $$

And now it is exactly the book answer.