1. We are given the trinomial $$10r^3s + 35rs^4 - 100rs$$ and asked to factor it, which represents the amount paid by Ms. Umali for bananas. 2. We will find the polynomial representing the shaded area in the figure: the area of the large circle minus the combined area of the six smaller circles inside it. ### Problem 1: Factoring the Trinomial 1. Identify the greatest common factor (GCF) of all terms: Each term contains $r$ and $s$, and the coefficients are 10, 35, and -100. The GCF of 10, 35, and 100 is 5. 2. Factor out the GCF: $$10r^3s + 35rs^4 - 100rs = 5rs(2r^2 + 7s^3 - 20)$$ 3. The factored form is: $$5rs(2r^2 + 7s^3 - 20)$$ This tells us the weight of bananas bought and the price per kilo relate as factors: one factor is $5rs$ representing common quantities, and the other $2r^2 + 7s^3 - 20$ could be price or weight components. ### Problem 2: Factoring the Polynomial for the Shaded Area 1. The large circle has radius $R$, so its area is: $$A_{large} = \pi R^2$$ 2. There are six small circles each with radius $r$, so combined area is: $$A_{small} = 6 \times \pi r^2 = 6\pi r^2$$ 3. The shaded area is the difference: $$A_{shaded} = A_{large} - A_{small} = \pi R^2 - 6\pi r^2$$ 4. Factor out common term $\pi$: $$A_{shaded} = \pi (R^2 - 6r^2)$$ 5. Recognize $R^2 - 6r^2$ as a difference of squares type but since 6 is not a perfect square, factoring stops here. Final factored polynomial representing the shaded area is: $$\pi (R^2 - 6r^2)$$