ME The expression \( \pi(r-3)^{2} \) represents the area covered by the hour hand a clock in one rotation, where \( r \) is the radius of the entire clock. Write a olynomial that represents the area covered by the hour hand in one rotation.

To express the area covered by the hour hand of a clock in one rotation as a polynomial, let's start with the given expression: \[ \text{Area} = \pi (r - 3)^2 \] Here, \( r \) is the radius of the entire clock, and \( \pi (r - 3)^2 \) represents the area of a circle with radius \( r - 3 \). To convert this into a polynomial, we'll expand the binomial \((r - 3)^2\): \[ (r - 3)^2 = r^2 - 6r + 9 \] Now, multiply each term by \( \pi \) to get the area: \[ \text{Area} = \pi (r^2 - 6r + 9) \] Distributing \( \pi \) across the terms gives: \[ \text{Area} = \pi r^2 - 6\pi r + 9\pi \] **Polynomial Representation:** \[ \text{Area}(r) = \pi r^2 - 6\pi r + 9\pi \] This polynomial accurately represents the area covered by the hour hand of the clock in one full rotation. **Final Answer:** After expanding, the polynomial is π r² minus six π r plus nine π. Thus, Area(r) = π r² − 6πr + 9π