The function provided in the image is:

$z(x, y) = y^2 \cdot xy - x \cdot y^2$

This expression can be simplified. Let's break down the terms and see if any simplifications can be made:

1. Expand the terms: $z(x, y) = y^2 \cdot x \cdot y - x \cdot y^2$

2. Notice that both terms contain $x \cdot y^2$. Thus, we can factor this term out: $z(x, y) = x \cdot y^2 (y - 1)$

So the simplified form of the function is:

$z(x, y) = x \cdot y^2 (y - 1)$

Would you like further details on this problem or its applications? Here are a few related questions to deepen your understanding:

1. How would you find the partial derivatives of $z(x, y)$ with respect to $x$ and $y$?
2. What are the critical points of $z(x, y)$, and how can you find them?
3. How does this function behave for different values of $x$ and $y$?
4. How can we interpret this function geometrically in three dimensions?
5. What happens if we change the function to $z(x, y) = y^3 \cdot x - x \cdot y^3$?

Tip: Simplifying expressions by factoring out common terms can often make further analysis, such as finding derivatives or solving equations, much easier.