For the function \( f(x)=x^{8}-7 x^{4}+5 x-5 \), state: a) the maximum number of real zeros that the function can have; b) the maximum number of \( x \)-intercepts that the graph of the function can have; and c) the maximum number of turning points that the graph of the function can have. The function \( f(x)=x^{8}-7 x^{4}+5 x-5 \) has a maximum of \( \square \) real zeros.

The polynomial \( f(x) = x^8 - 7x^4 + 5x - 5 \) is of degree 8, which means it can have a maximum of 8 real zeros. For polynomial functions, the number of real zeros cannot exceed the degree of the polynomial. Thus, this function can have a maximum of 8 real zeros. The graph of the function can also have a maximum of 8 \( x \)-intercepts since each zero corresponds to an \( x \)-intercept. Since polynomial functions are continuous, each time the function crosses the x-axis, it represents a real zero, which means that in the best-case scenario, each of these could be distinct and thus show up as separate intercepts. When it comes to turning points, a polynomial of degree \( n \) can have a maximum of \( n - 1 \) turning points. Therefore, for this function, the maximum number of turning points possible is \( 8 - 1 = 7 \). Taking all these into consideration, we can conclude: The function \( f(x) = x^8 - 7x^4 + 5x - 5 \) has a maximum of 8 real zeros.