By the Rational Zeroes Theorem (RZT), if a polynomial has only integer coefficients, then any rational zero must be the positive or negative quotient of a factor of the constant and a factor of the coefficient of greatest degree. These integers are, respectively, 24, which as as its factors 1, 2, 3, 4, 6, 8, 12, and 24, and 4, which has as its factors 1, 2, and 4.

The complete set of quotients of factors of the former and factors of the latter is derived by dividing each element of  by each element of . The resulting set is 

,

so any rational zero must be an element of this set.  is not an element of this set, so by the RZT, it cannot be a zero of the polynomial.

The easiest way to answer this question is arguably as follows:

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 two such terms - the fifth-degree and first-degree (linear) term, so alternating coefficient sum is  

.

It follows that  is divisible by .

The easiest way to answer this question is arguably as follows:

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

as its coefficient sum, so  is not divisible by .

One way to answer this question is as follows:

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

as its coefficient sum, so  is indeed divisible by .

If a polynomial has only integer coefficients, it is possible for its zeroes to be real or imaginary; however, by the Complex Conjugate Roots Theorem, any imaginary zeroes occur in conjugate pairs. For example, if  is a zero, so is . Therefore, such a polynomial must have an even number of imaginary zeroes, distinct or otherwise. But by the Fundamental Theorem of Algebra, a polynomial of degree 7 has 7 zeroes, distinct or otherwise. Therefore, the polynomial has six distinct imaginary zeroes at most.

By the Fundamental Theorem of Algebra, a polynomial of degree  has  zeroes that may or may not be distinct. By the Conjugate Zeroes Theorem, if the polynomial has only rational coefficients, then its imaginary zeroes occur in conjugate pairs, so there must be an even number of them. If the polynomial has degree six, then there are three pairs, and they may be distinct, for a total of six distinct imaginary zeroes. 

In fact, we can construct an example of a polynomial with six distinct zeroes: , as follows:

If a polynomial has  as a zero, then its factorization, if taken down to linear binomials, includes the factor .

Examine the polynomial 

,

which has as its set of zeroes  - six distinct imaginary numbers. Applying the sum/difference pattern of binomial multiplication three times:

This polynomial has, by construction, integer coefficients and six distinct imaginary zeroes.

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.