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 8 real zeros. The function $$ f(x)=x^{8}-7 x^{4}+5 x-5 $$ has a maximum of 8 x-intercepts. The function $$ f(x)=x^{8}-7 x^{4}+5 x-5 $$ has a maximum of

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We start by noting that $$ f(x)=x^{8}-7x^{4}+5x-5 $$ is a polynomial of degree $$ 8 $$.

1. Since the degree is $$ 8 $$, the Fundamental Theorem of Algebra ensures that there are exactly $$ 8 $$ complex zeros (counting multiplicities). Therefore, the maximum number of real zeros (when all zeros are real and distinct) is  
   $$
   8.
   $$

2. An $$ x $$-intercept occurs at a real zero of the function. Thus, if all $$ 8 $$ zeros are real and distinct, the graph will have  
   $$
   8\text{ x-intercepts}.
   $$

3. The maximum number of turning points of a polynomial of degree $$ n $$ is $$ n-1 $$. Hence, for a degree $$ 8 $$ polynomial, the maximum number of turning points is  
   $$
   8-1=7.
   $$

Thus, the answers are:  
a) $$ 8 $$ real zeros;  
b) $$ 8 $$ $$ x $$-intercepts;  
c) $$ 7 $$ turning points.
a) The function can have up to 8 real zeros. b) The graph can have up to 8 $$ x $$-intercepts. c) The graph can have up to 7 turning points.

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