To add the expressions [tex]\( p(p - q) \)[/tex], [tex]\( q(q - r) \)[/tex], and [tex]\( r(r - p) \)[/tex], we should first expand each expression and then sum the results together. Here are the steps:
1. Expand each expression:
[tex]\[
p(p - q) = p^2 - pq
\][/tex]
[tex]\[
q(q - r) = q^2 - qr
\][/tex]
[tex]\[
r(r - p) = r^2 - rp
\][/tex]
2. Sum the expanded expressions:
[tex]\[
p^2 - pq + q^2 - qr + r^2 - rp
\][/tex]
3. Combine like terms:
- Collect all the [tex]\( p^2, q^2, \)[/tex] and [tex]\( r^2 \)[/tex] terms:
[tex]\[
p^2 + q^2 + r^2
\][/tex]
- Collect all the terms involving [tex]\( pq \)[/tex], [tex]\( qr \)[/tex], and [tex]\( rp \)[/tex]:
[tex]\[
-pq - qr - rp
\][/tex]
4. Write the final expression:
Bringing everything together, the sum of the expressions [tex]\( p(p - q) \)[/tex], [tex]\( q(q - r) \)[/tex], and [tex]\( r(r - p) \)[/tex] is:
[tex]\[
p^2 + q^2 + r^2 - pq - qr - rp
\][/tex]
Thus, the step-by-step solution for summing the given expressions is as follows:
[tex]\[
p(p - q) + q(q - r) + r(r - p) = p^2 + q^2 + r^2 - pq - qr - rp
\][/tex]
