Final answer:
The long run behavior of the function f(p) is governed by the term p^7, which means as p becomes very large or small, f(p) will increase without bound in the positive or negative direction, respectively.
Step-by-step explanation:
The long run behavior of the function f(p) = p^7 + 3p^6 + 2p^5 + 5 involves analyzing what happens as the variable p gets very large (either positively or negatively).
To determine this behavior, we consider the term with the highest power of p since it will be the most influential as p increases or decreases without bound. In this case, the highest power of p is 7 in the first term p^7.
As p becomes very large, p^7 will dominate the other terms in the function; therefore, the long run behavior of this polynomial function is that it will increase without bound as p becomes very large or very small in the positive or negative direction, respectively.
This is because the leading term p^7 dictates the end behavior of the polynomial, and since the exponent is odd and positive, f(p) will approach positive infinity as p approaches positive infinity and negative infinity as p approaches negative infinity.