Let's solve the problem based on the given diagram.

Analyzing the figure:

1. Total area of the large square: The side length of the large square is $z$. Thus, the total area is: $A_{\text{total}} = z^2$

2. Non-shaded (white) region:

   • Top rectangle: The dimensions are $z \times 2$, so its area is: $A_{\text{top rectangle}} = z \cdot 2 = 2z$

   • Right rectangle: The dimensions are $16 \times 2$, so its area is: $A_{\text{right rectangle}} = 16 \cdot 2 = 32$

   • Total area of the non-shaded region: $A_{\text{non-shaded}} = A_{\text{top rectangle}} + A_{\text{right rectangle}} = 2z + 32$

3. Shaded area: The shaded area is the difference between the total area and the non-shaded region: $A_{\text{shaded}} = A_{\text{total}} - A_{\text{non-shaded}}$ Substituting the values: $A_{\text{shaded}} = z^2 - (2z + 32)$

   Simplify the expression: $A_{\text{shaded}} = z^2 - 2z - 32$

Final Answer:

The polynomial for the shaded area is: $z^2 - 2z - 32$

Would you like further explanation or clarification? Let me know!

Related Questions:

1. How do you find the total area of a composite shape?
2. What is the importance of subtracting non-overlapping areas?
3. How can we verify the polynomial is correct?
4. What happens if the dimensions of the rectangles change?
5. How do we classify polynomials based on their terms and degrees?

Tip:

Always ensure each component of a figure is labeled and accounted for to avoid errors when calculating areas!