Question 1189813
Here's a possible roller coaster design using a polynomial function, addressing all the requirements:

**1. Roots/x-intercepts:**

I've chosen the following x-intercepts, keeping in mind the 120-second requirement and avoiding 1 and 2:

*   x = 0 (starting point)
*   x = 10 (first hill)
*   x = 40 (first valley/tunnel entrance)
*   x = 40 (double root - tunnel bottom)
*   x = 60 (tunnel exit)
*   x = 90 (second hill)
*   x = 110 (second valley)
*   x = 120 (ride's end)

**2. Equation:**

To create the equation, we'll use the factored form of a polynomial.  Each root corresponds to a factor:

f(x) = A * x * (x - 10) * (x - 40)² * (x - 60) * (x - 90) * (x - 110) * (x - 120)

*   **A:**  This is a scaling factor. We'll determine it using the starting height of 250 feet.

Since f(0) = 250:

250 = A * 0 * (-10) * (-40)² * (-60) * (-90) * (-110) * (-120)

Notice that if we plug in zero we will get zero, so we need to change the formula to fit the initial condition, we will add 250 to the formula.

f(x) = A * x * (x - 10) * (x - 40)² * (x - 60) * (x - 90) * (x - 110) * (x - 120) + 250

Now plug in zero:

250 = A * 0 * (-10) * (-40)² * (-60) * (-90) * (-110) * (-120) + 250

250 = 250

We still need to find A, to do this we need to select a point in the graph and plug it in, for example f(5) = 300

300 = A * 5 * (-5) * (-35)² * (-55) * (-85) * (-105) * (-115) + 250

50 = A * 5 * (-5) * (-35)² * (-55) * (-85) * (-105) * (-115)

Solving for A:

A ≈ 1.357e-11

Therefore, the final equation is:

f(x) = 1.357e-11 * x * (x - 10) * (x - 40)² * (x - 60) * (x - 90) * (x - 110) * (x - 120) + 250

**Explanation:**

*   Each factor (x - root) makes the function equal to zero at that root, creating the x-intercepts.
*   The double root at x = 40 makes the graph "kiss" the x-axis at that point, creating the tunnel.
*   The scaling factor A controls the overall shape and height of the roller coaster. We solve for A using the initial condition f(0) = 250.

**Important Notes:**

*   This equation will likely produce very large numbers due to the high powers of x. Graphing software might struggle with it directly.  You might need to adjust the scaling factor *A* or use a different approach to get a visually pleasing graph.
*   You can adjust the positions and multiplicities of the roots to fine-tune the shape of your roller coaster.
*   Be sure to use graphing software (like Desmos or Wolfram Alpha) to visualize the function and make sure it meets all the requirements (3 relative extrema, below ground level, 120 seconds long).  You may have to iterate a few times to get the shape you want.