Given the function $f(x) = x^7 - x^2 + 8$, let’s determine the following properties:

1. Maximum Number of Real Zeros

The maximum number of real zeros of a polynomial is equal to its degree. Here, the degree of $f(x)$ is $7$ (the highest power of $x$). Therefore, the maximum number of real zeros is:

$7$

2. Maximum Number of $x$-Intercepts

The $x$-intercepts of a function are the points where $f(x) = 0$. Since real zeros correspond to $x$-intercepts, the maximum number of $x$-intercepts is also:

$7$

3. Maximum Number of Turning Points

The maximum number of turning points of a polynomial function is given by $n - 1$, where $n$ is the degree of the polynomial. For $f(x)$, the degree is $7$, so the maximum number of turning points is:

$7 - 1 = 6$

Summary

• Maximum number of real zeros: $7$
• Maximum number of $x$-intercepts: $7$
• Maximum number of turning points: $6$

Would you like further details on any of these concepts?

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Here are some related questions for further exploration:

1. How do you determine the possible number of complex roots for a polynomial?
2. Why is the maximum number of turning points always one less than the degree of the polynomial?
3. How do we find the actual turning points of a polynomial function?
4. Can a polynomial with an odd degree ever have no real zeros?
5. How can the behavior of the end terms affect the shape of the polynomial graph?

Tip: Remember, a polynomial function of degree $n$ has at most $n$ real zeros, but it may have fewer depending on the nature of its roots.