Let . By a corollary of the Factor Theorem,  is divisible by  if and only if the alternating sum of its coefficients (accounting for minus symbols) is 0.

To find this alternating sum, it is necessary to reverse the symbol before all terms of odd degree. In , there are no such terms, (the degree of the constant is 0), so the alternating coefficient sum is  

,

and  is divisible by .

The highest point of the ball is the vertex of the ball's parabolic path, so to find the number of seconds  that is takes to reach this point, it is necessary to find the vertex of the parabola of the graph of the function

The parabola of the graph of 

has as its ordinate, or -coordinate,

,

so, setting ,

,

which rounds to 2.3 seconds. This is the time that it takes the ball to reach the highest point of its path.

When the baseball hits the ground, its height is 0; therefore, we are looking for  such that 

,

or

This equation can most easily be solved using the quadratic formula. If 

,

then 

Setting :

One possible answer is 

We throw this out, since we cannot have "negative time".

The other is 

This is positive, so we accept this answer. The ball hits the ground in about 6.6 seconds.

Using the FOIL (first, outer, inner, last) method, you can expand the polynomial to get 

first: 

outer: 

inner: 

lasts: 

From here, combine the like terms.

In order to find the roots, we must factor the equation.

The factors of this equation are  and .

Setting those two equal to zero, we get  and .

To find where the graph crosses the horizontal axis, we need to set the function equal to 0, since the value at any point along the axis is always zero.

To find the possible rational zeroes of a polynomial, use the rational zeroes theorem:

Our constant is 10, and our leading coefficient is 1. So here are our possible roots:

Let's try all of them and see if they work! We're going to substitute each value in for using synthetic substitution. We'll try -1 first.

Looks like that worked! We got 0 as our final answer after synthetic substitution. What's left in the bottom row helps us factor down a little farther:

We keep doing this process until is completely factored:

Thus, crosses the axis at .

crosses the axis when equals 0. So, substitute in 0 for :

We have the following answer choices.

The first equation is a cubic function, which produces a function similar to the graph. The second equation is quadratic and thus, a parabola. The graph does not look like a prabola, so the 2nd equation will be incorrect. The third equation describes a line, but the graph is not linear; the third equation is incorrect. The fourth equation is incorrect because it is an exponential, and the graph is not an exponential. So that leaves the first equation as the best possible choice.

The highest exponent of the variable term is two (). This tells that this function is quadratic, meaning that it is a parabola.

The graph below will be the answer, as it shows a parabolic curve.

When determining the maximum number of turns a polynomial function might have, one must remember: 

Max Number of Turns for Polynomial Function = degree - 1

First, we must find the degree, in order to determine the degree we must put the polynomial in standard form, which means organize the exponents in decreasing order: 

Now that f(x) is in standard form, the degree is the largest exponent, which is 8. 

We now plug this into the above: 

Max Number of Turns for Polynomial Function = degree - 1

Max Number of Turns for Polynomial Function = 8 - 1 

which is 7. 

The correct answer is 7.