Expressing an algebraic function as a single expression I have the following question in my text book:
Express as a single fraction
$$ \frac{3}{x-4} - \frac{2}{(x-4)^2} $$
The answer the book gives is this:
$$ \frac{3x-14}{(x-4)^2} $$
I understand how they get to this through these steps:
$$ \frac{3(x-4)-2}{(x-4)^2} = \frac{3x-12-2}{(x-4)^2} $$
My question is why can you not cancel the $$ (x-4) $$ instead like this:
$$ \frac{3(x-4)-2}{(x-4)^2} = \frac{3-2}{(x-4)} = \frac{1}{(x-4)} $$

It is never allowed to cancel across a plus or minus in a fractional expression.  If it were possible, strange things would happen.  Consider your expression when $x = 6$:
$$
  \frac{3(x-4) - 2}{(x-4)^2} = \frac{3(6-4)-2}{(6-4)^2} = \frac{3(2) - 2}{2^2}
= \frac{4}{4} = 1.
$$
But if you tried canceling the $x-4$ first:
$$\frac{3(x-4) - 2}{(x-4)^2} \stackrel{?}{=} \frac{3-2}{(x-4)} = \frac{1}{x-4} = \frac{1}{6-4} =\frac{1}{2}.
$$
Not the same, right?
Hope this helps!
