During the first 5 seconds of an amusement park ride, the function f(x)=x5–5.5x4+2.5x3+20x2–26x+8 represents the height (in feet) of the ride after x seconds. How long is the ride at or below the ground during the first 5 seconds?
In this particular case, f(x) represents the height of the ride. Therefore, if the ride is at the ground level, then f(x) = 0. Therefore, we should approach this problem by letting f(x) = 0, and begin the process of factoring to find all possible x-intercepts. Because the degree is ODD (leading exponent is a 5), it must have at least ONE x-intercept (but likely has more - it could have 1, 3, or 5 real x-intercepts).
To begin: Let x5 – 5.5x4 + 2.5x3 + 20x2 – 26x + 8 = 0
> Factor this polynomial to find the individual x-intercepts.
> Find the difference in the x-intercepts where the height of the graph is less than/equal to 0 between times of x = 0 and x = 5.
I will leave the factoring portion up to you to solve.
Tom K.
Multiplying through by 2 may help. You can then use the rational root theorem. Multiplying by 2 also helps if you want to use synthetic division. It turns out that this polynomial has 5 rational roots. You can plot, also, to find them.08/27/21