To simplify the expression [tex]\(\left(-8 p^3 - 11 p^2 q \right) + \left(p q^2 - 5 q^3 \right) - \left(-6 p^2 q + p q^2 \right)\)[/tex], let's break it down step by step:
1. Write out the expression:
[tex]\[
\left(-8 p^3 - 11 p^2 q \right) + \left(p q^2 - 5 q^3 \right) - \left(-6 p^2 q + p q^2 \right)
\][/tex]
2. Distribute the negative sign through the third term:
[tex]\[
-\left(-6 p^2 q + p q^2 \right) = 6 p^2 q - p q^2
\][/tex]
Now the expression is:
[tex]\[
\left(-8 p^3 - 11 p^2 q \right) + \left(p q^2 - 5 q^3 \right) + \left(6 p^2 q - p q^2 \right)
\][/tex]
3. Combine like terms:
- For [tex]\(p^3\)[/tex]:
[tex]\[
-8 p^3
\][/tex]
- For [tex]\(p^2 q\)[/tex]:
[tex]\[
-11 p^2 q + 6 p^2 q = -5 p^2 q
\][/tex]
- For [tex]\(p q^2\)[/tex]:
[tex]\[
p q^2 - p q^2 = 0
\][/tex]
- For [tex]\(q^3\)[/tex]:
[tex]\[
-5 q^3
\][/tex]
4. Putting it all together:
[tex]\[
-8 p^3 - 5 p^2 q + 0 - 5 q^3
\][/tex]
Simplify by removing the zero term:
[tex]\[
-8 p^3 - 5 p^2 q - 5 q^3
\][/tex]
So, the simplified expression is:
[tex]\[
-8 p^3 - 5 p^2 q - 5 q^3
\][/tex]
