# Algebra II 4

As one of the new roller coaster engineers, you have been tasked with developing a roller coaster that will intertwine with existing Oakville Lake Amusement Park structures. For one of the more thrilling sections, the roller coaster will dive down in-between buildings, plummet underground, pop back up, and coast over a hill before shooting back underground. There must be three distinct points where the roller coaster crosses the x–axis. Precise measurements and attention to detail are very important.

First, here is the existing map of current structures. It is important that the roller coaster does not go through the foundation of any of these structures. 1st point: ___6___

2nd point:___-2___

3rd point: ___-7___

1. Using the points above as zeros, construct the polynomial function, f(x), that will be the path of your roller coaster. Show all of your work.

Answer: To create the polynomial function, we would need to find the three roots to create factors. The roots would be the x-coordinates, 6, -2, and -7 so the factors are x-6, x+2 and x+7. Now multiple x-6 and x+2 to get x^2-4x-12. Take that answer and multiply it by the third factor,x+7 and it would result with x^3+x^2-28x-12x-84. Combine like terms and the polynomial function is f(x) = x^3+x^2-40x-84. 2. Using both fundamental Theorem and Descartes` rule of signs, prove to the construction foreman that your function matches your graph. Use complete sentences.

Answer: Using Descartes’ Rule of Signs can determine the possible numbers of solutions to the equation. By looking at f(x) = x^3+x^2-40x-84, you can see that there is one sign change meaning that there is one positive root. Now look at f(-x) version of the previous equation: f(-x)=-x^3+x^2+40x-84, there are two sign changes meaning there are two negative roots. Using the Descartes’ Rules of Signs on the equation f(x) = x^3+x^2-40x-84, told us that there are one positive root and two negative roots....

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