Final answer:
The degree of the polynomial is 9. The roots of the polynomial are x = 4 (multiplicity 7), x = 0 (multiplicity 5), and x = 0 (multiplicity 9). The simplified expression is p(x) = (x^7 - 28x^6 + 294x^5 - 1680x^4 + 5292x^3 - 9604x^2 + 8064x) * (x^5) * (ax^9). The end behavior of the polynomial is determined by the leading term, which is ax^9.
Step-by-step explanation:
a) The degree of a polynomial is determined by the highest exponent of the variable. In this case, the highest exponent is 9, so the degree of the polynomial is 9.
b) To find the roots of the polynomial, we set the polynomial equal to zero and solve for x. Since we have several factors, we also need to consider the multiplicities of each factor. From the given polynomial, we have the following roots: x = 4 (multiplicity 7), x = 0 (multiplicity 5), and x = 0 (multiplicity 9).
c) To simplify the expression, we can expand each factor and combine like terms. The expanded form of the polynomial is p(x) = (x^7 - 28x^6 + 294x^5 - 1680x^4 + 5292x^3 - 9604x^2 + 8064x) * (x^5) * (ax^9).
d) The end behavior of the polynomial is determined by the leading term, which is the term with the highest degree. Since the leading term is ax^9, the end behavior of the polynomial is the same as that of a ninth-degree polynomial. If the leading coefficient a is positive, then the polynomial increases without bound on both ends. If a is negative, then the polynomial decreases without bound on both ends.