Let's simplify the expression [tex]\(\frac{1}{x-y} - \frac{y}{xy + y^2}\)[/tex].
1. Identify common denominators:
The denominators of our two fractions are [tex]\(x - y\)[/tex] and [tex]\(xy + y^2\)[/tex].
2. Rewrite the second denominator:
Observe that [tex]\(xy + y^2\)[/tex] can be factored:
[tex]\[
xy + y^2 = y(x + y)
\][/tex]
Therefore, the expression can be rewritten as:
[tex]\[
\frac{1}{x - y} - \frac{y}{y(x + y)} = \frac{1}{x - y} - \frac{1}{x + y}
\][/tex]
3. Find a common denominator:
The common denominator for [tex]\(x-y\)[/tex] and [tex]\(x+y\)[/tex] is [tex]\((x-y)(x+y)\)[/tex].
4. Rewrite each fraction:
Rewrite each fraction with the common denominator:
[tex]\[
\frac{1}{x - y} = \frac{x + y}{(x - y)(x + y)}
\][/tex]
[tex]\[
\frac{1}{x + y} = \frac{x - y}{(x - y)(x + y)}
\][/tex]
5. Subtract the fractions:
Now subtract the two fractions:
[tex]\[
\frac{x + y}{(x - y)(x + y)} - \frac{x - y}{(x - y)(x + y)}
\][/tex]
Combining the numerators:
[tex]\[
\frac{(x + y) - (x - y)}{(x - y)(x + y)} = \frac{x + y - x + y}{(x - y)(x + y)}
\][/tex]
Simplify the numerator:
[tex]\[
\frac{2y}{(x - y)(x + y)}
\][/tex]
So the simplified form of the expression [tex]\(\frac{1}{x-y} - \frac{y}{xy + y^2}\)[/tex] is:
[tex]\[
\boxed{\frac{2y}{(x - y)(x + y)}}
\][/tex]
