To simplify the expression
[tex]\[
\frac{p^2 + pq + q^2}{p + q} - \frac{p^2 - pq + q^2}{p + q},
\][/tex]
we proceed as follows:
1. Combine the fractions over a common denominator:
Both fractions have the same denominator, [tex]\( p + q \)[/tex]. Therefore, we can combine them into a single fraction:
[tex]\[
\frac{(p^2 + pq + q^2) - (p^2 - pq + q^2)}{p + q}.
\][/tex]
2. Simplify the numerator by distributing and combining like terms:
Let's expand the numerator:
[tex]\[
(p^2 + pq + q^2) - (p^2 - pq + q^2).
\][/tex]
Distribute the negative sign through the terms inside the second parenthesis:
[tex]\[
p^2 + pq + q^2 - p^2 + pq - q^2.
\][/tex]
Combine like terms. Notice that [tex]\( p^2 - p^2 \)[/tex] and [tex]\( q^2 - q^2 \)[/tex] cancel out:
[tex]\[
pq + pq = 2pq.
\][/tex]
So, the numerator simplifies to [tex]\( 2pq \)[/tex].
3. Form the simplified fraction:
Now, we have the simplified numerator over the common denominator:
[tex]\[
\frac{2pq}{p + q}.
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\frac{2pq}{p + q}.
\][/tex]
