Answer:
To verify the zeros of the function \(f(x) = -x^3 + 7x^2 - 7x - 15\), we substitute each zero into the function and check if the result is zero.
For \(x = 5\):
\[f(5) = -(5)^3 + 7(5)^2 - 7(5) - 15\]
\[= -125 + 175 - 35 - 15\]
\[= 0.\]
For \(x = 3\):
\[f(3) = -(3)^3 + 7(3)^2 - 7(3) - 15\]
\[= -27 + 63 - 21 - 15\]
\[= 0.\]
For \(x = -1\):
\[f(-1) = -(-1)^3 + 7(-1)^2 - 7(-1) - 15\]
\[= -(-1) + 7 - (-7) - 15\]
\[= 0.\]
Since the function evaluates to zero for each of the given values, 5, 3, and -1, these are indeed the zeros of the function \(f(x)\). Now, to describe the end behavior of the function, we examine the leading term of the polynomial. The leading term is \(-x^3\), which implies that as \(x\) approaches positive or negative infinity, the function's value also approaches negative infinity. Thus, the end behavior of the function \(f(x)\) is that it decreases without bound as \(x\) becomes very large positively or negatively