For the function​ f(x), find the maximum number of real​ zeros, the maximum number of​ x-intercepts, and the maximum number of turning points that the function can have.

​f(x)equals=x Superscript 6 Baseline minus x squared plus 9

Given the function \( f(x) = x^6 - x^2 + 9 \), we can determine the following:

### 1. **Maximum Number of Real Zeros**:
The maximum number of real zeros corresponds to the degree of the polynomial, which is the highest power of \( x \). Since the degree of \( f(x) \) is 6, the function can have up to **6 real zeros**. However, we must note that not all of these zeros must be real; some could be complex.

### 2. **Maximum Number of \( x \)-Intercepts**:
The maximum number of \( x \)-intercepts corresponds to the number of real zeros of the function. As mentioned, the function can have up to **6 \( x \)-intercepts**, assuming all zeros are real.

### 3. **Maximum Number of Turning Points**:
The maximum number of turning points (local maxima and minima) is given by \( n - 1 \), where \( n \) is the degree of the polynomial. For \( f(x) = x^6 - x^2 + 9 \), the degree is 6, so the function can have at most \( 6 - 1 = 5 \) turning points.

### Summary:
1. **Maximum number of real zeros**: 6
2. **Maximum number of \( x \)-intercepts**: 6
3. **Maximum number of turning points**: 5

Would you like more details or further clarifications on this?

### 5 Related Questions:
1. How do we determine if a polynomial function has complex roots?
2. Can a polynomial with real coefficients have an odd number of turning points?
3. What happens when all the zeros of a polynomial are complex?
4. How do the multiplicities of roots affect the graph of a polynomial?
5. What is the role of the leading coefficient in determining the end behavior of a polynomial?

**Tip**: The number of turning points is always one less than the degree of the polynomial, but it doesn't guarantee that all turning points are real.

For the function​ f(x), find the maximum number of real​ zeros, the maximum number of​ x-intercepts, and the maximum number of turning points that the function can have.
f(x) = x^6 - x^2 + 9

Learn how to find the maximum number of real zeros, x-intercepts, and turning points for the polynomial function f(x) = x^6 - x^2 + 9. This detailed explanation covers key algebraic concepts, including polynomial degrees and the Fundamental Theorem of Algebra, suitable for students in Grades 10-12.

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