Answer:
Part A:
- There is a zero at x = 7
- There is a zero at x = -2
- There is a zero at x = 3
Part B:
As x approaches +∞, f(x) approaches +∞
As x approaches -∞, f(x) approaches -∞
Step-by-step explanation:
To check if a point is a zero on a function, plug that point in for x. If the function(f(x)) equals 0 at that point, then it is a zero. For end behavior, look at the x term with the largest degree. This term grows at a quicker rate than the other terms and will dictate the overall end behavior(as the function approaches infinity from both sides).
Solving:
[tex]\section*{Part A}To determine if \( 7, -2, \) and \( 3 \) are zeros of \( f(x) \), we need to evaluate \( f(x) \) at these values. If \( f(x) = 0 \) for any of these values, then that value is a zero of the function.\subsubsection*{ \( f(7) \):}\[f(7) = 7^3 - 8 \cdot 7^2 + 7 + 42\]\[= 343 - 8 \cdot 49 + 7 + 42\]\[= 343 - 392 + 7 + 42\]\[= 343 - 392 + 49\]\[= 343 - 343\]\[=\boxed{0}\\\\\]\Since \( f(7) = 0 \), \( x = 7 \) is a zero of \( f(x) \).[/tex]
[tex]\subsubsection*{\( f(-2):\)}\[f(-2) = (-2)^3 - 8 \cdot (-2)^2 + (-2) + 42\]\[= -8 - 8 \cdot 4 - 2 + 42\]\[= -8 - 32 - 2 + 42\]\[= -40 - 2 + 42\]\[= -42 + 42\]\[\boxed{= 0}\]Since \( f(-2) = 0 \), \( x = -2 \) is a zero of \( f(x) \).[/tex]
[tex]\subsubsection*{ \( f(3) \):}\[f(3) = 3^3 - 8 \cdot 3^2 + 3 + 42\]\[= 27 - 8 \cdot 9 + 3 + 42\]\[= 27 - 72 + 3 + 42\]\[= 27 - 72 + 45\]\[= -45 + 45\]\[\boxed{= 0}\]Since \( f(3) = 0 \), \( x = 3 \) is a zero of \( f(x) \).[/tex]
[tex]\hrulefill[/tex]
[tex]\subsection*{Part B}The end behavior of a polynomial function is determined by its highest degree. For \( f(x) = x^3 - 8x^2 + x + 42 \), the leading term is \( \boxed{x^3 }\).\begin{itemize} \item As \( x \) approaches \( +\infty \), \( x^3 \) will dominate and \( f(x) \) will approach \( +\infty \). \item As \( x \) approaches \( -\infty \), \( x^3 \) will dominate and \( f(x) \) will approach \( -\infty \).\end{itemize}[/tex]
[tex]\text{Therefore, the end behavior of \( f(x) \) is:}\\\begin{itemize} \item As \( x \to +\infty \), \( f(x) \to +\infty \). \item As \( x \to -\infty \), \( f(x) \to -\infty \).\end{itemize}[/tex]
[tex]\hrulefill[/tex]
