Polynomials are everywhere in the world around you! From roller coaster and bridge design, to economic growth patterns, to supermarket inventory management and more, you can find polynomials everywhere you look! Since polynomials can be used to model so many phenomena in our world, mathematicians have developed several theorems to help us. Two of these theorems are listed below. The Factor Theorem A polynomial P(x) has a factor (x a) if and only if P(a) = 0. Think about these theorems as you consider the application of polynomials below. An animal sanctuary is constructing a large tank for its exotic fish collection. The tank will be a rectangular prism that is set into a large wall, and it will feature two, equally-sized hemispheres, one on each side of the tank. Visitors will be able =8 to sit inside these hemispheres so they can feel | like they are inside the tank with the fish. The polynomial (2x 30) represents the length of the new tank, the polynomial (2x 20) represents the width of the new tank, and the polynomial (2x 60) represents the height of the new tank. Use the information given to explore some of the mathematical concepts you have practiced so far by answering the questions below. 1. Find a polynomial that represents volume of the fish tank. Explain how you used the properties of exponents to determine your expression. HINT: The formula for the volume of a rectangular prism is V = [wh. 2. The volume of each hemisphere is represented by the polynomial x* 70x2 + 360x 1800. Explain how to rewrite your answer for question 1 to reflect the volume of the fish tank after the hemispheres are installed. Then carry out your plan. Show your work. 3. Show that the binomial that represents the length of the fish tank is a factor of the polynomial you wrote in question 1. 4. Is the binomial that represents the length of the fish tank a factor of the polynomial that represents the volume of the fish tank after the hemispheres are installed? Support your answer mathematically. 5. The unthinkable happened! One of the hemispheres in the tank sprung a leak and had to be replaced. It was replaced by a new hemisphere whose volume can be represented by the polynomial 2% 65x? + 420 x 1710. What is the volume of the fish tank after this hemisphere is installed? HINT: The other original hemisphere did not experience any problems and is still installed in the tank