1. Polynomial Rollercoaster Project
Mrs. Chernowski Pre-Calculus Chris Murphy

2. Key: t (x-cord) in terms of minutes. h(t) (y-cord) in terms of feet.
Requirements:
At least 3 relative maxima and/or minima
The ride length must be at least 4 minutes
The coaster ride starts at 250 feet
The ride dives below the ground into a tunnel at least once
Key: t (x-cord) in terms of minutes. h(t) (y-cord) in terms of feet.

3. Real roots:
1.25 (1 ¼) → (x ), 2 → (x - 2), 3 → (x - 3), 4.5 (4 ½) → (x - 4.5)
Double root: - (When the graph touches the x-axis once producing the same root twice.)
3 → (x - 3)2
Imaginary root:
1 + 2i, 1 - 2i → (x2 – 2x + 5)

4. Factoring the Polynomial
Since the visible roots on the graph are: - (real roots and double root)
1.25, 2, 3, and 4.5 this concludes to → #2.)x= 1.25, 2, 3, 4.5
Since the polynomial also requires an imaginary root I included a function that results in the inclusion of “i” (an imaginary number) by using a quadratic equation that cannot be simplified into a real number by the quadratic formula. This imaginary root was assured by selecting values that allow a negative value in the “square root section”:
Selected example quadratic equation: (x2-2x+5)
Selected quadratic formula values: a = 1, b = -2, c = 5
x= [-(-2) ± √(-2)2 – 4(1)(5)]/2 → (2 ± √-16)/2 → “1 ± 4i” (i = √-1)
x = 1 + 2i, 1 – 2i
Therefore, (x2-2x+5) = #2.) [x = 1 + 2i, 1 – 2i]

5. Factoring the Polynomial (continued)
In conclusion, all of the roots (real, double, and imaginary) can be stated as:
(x – 1 .25)(x – 2)(x – 3)(x – 4.5)(x2 – 2x + 5)
(x2-2x-1.25x+2.5)(x2-4.5x-3x+13.5)(x2-2x+5)
(x2-3.25x+2.5)(x2-7.5x+13.5)(x2-2x+5)
(x x x x+33.75)(x2-2x+5)
x x x x x x
#.3) Complete factored polynomial:
y = h(t) = x x x x x x
This polynomial function depicts the rollercoaster with a leading coefficient of 1, although it is not in standard form because it currently includes decimals.

6. Factoring the Polynomial Standard Form
Standard form: - The coefficients of each degree (including the constant) must be represented as integers (a number that does not contain decimals/fractions)
Since the polynomial equation was distinguished as:
y = h(t) = x x x x x x
The decimals of the coefficients and constant are then transformed into fractions to obtain the LCD (Least Common Denominator)
x6-12(¾)x5+66(⅞)x4-197(⅛)x3+360(⅞)x2-380(⅝)x+168(¾)
Since the LCD is 8, the function is multiplied as a whole by 8 to eliminate the fractions.
8 * [x6-12(¾)x5+66(⅞)x4-197(⅛)x3+360(⅞)x2-380(⅝)x+168(¾)]
8x6-102x5+535x4-1577x3+2887x2-3045x+1350
#4.) Polynomial in standard form:
h(t) = 8x6-102x5+535x4-1577x3+2887x2-3045x+1350

7. Dividing the Polynomial (Synthetic Division)
#.5) To verify the roots of the equation, synthetic division is applied to all roots:
Roots: 1.25, 2, 3, 4.5, 1+2i, 1-2i
1.25|
÷ ↓ | 2|
÷ ↓ 3|
÷ ↓ |
4.5|
÷ ↓ |
Since the remainder is 0, (x-1.25) is a factor of the entire polynomial.

8. Dividing the Polynomial (Synthetic Division) (continued)
#.5) Now the imaginary root(s) must be synthetically divided to verify it as a factor of the polynomial.
Imaginary roots: 1-2i, 1+2i → “(x2 – 2x + 5)”
The depressed polynomial (the coefficients above) from 1-2i is then used as the dividend when using 1+2i as the divisor (as shown below).

9. End Behavior of Function
Since the leading coefficient An has an even degree “6” and An > 0 the graph will rise to both the left and the right.
At the decreasing interval at the end of the graph represents:
y approaches -∞ as x approaches (+)∞
But there is a range and domain restriction that does not make this behavior continuous.

10. The Practical Range and Domain of the Graph
#8.) Practical domain:
0 ≤ x ≤ 4.5
[0, 4.5]
#9.) Practical range:
-50 ≤ y ≤ 350
[-50, 350]

11. Increasing, Decreasing, and Constant Intervals
#10.) Increasing intervals: (0 ≤ x ≤ 0.5), (1.5 ≤ x ≤ 2.5), (3 ≤ x ≤ 3.5) → (0, 0.5)U(1.5, 2.5)U(3,3.5)
Decreasing intervals: (0.5 ≤ x ≤ 1.5), (2.5 ≤ x ≤ 3), (3.75 ≤ x ≤ 4.5) → (0.5, 1.5)U(2.5, 3)U(3.75, 4.5)
Constant interval: (3.5 ≤ x ≤ 3.75) → (3.5, 3.75)