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Let's break down the polynomial \( f(x) = 5x^6 + 2x^5 + 1 - 4x^3 \) to answer the given questions:

1. **Degree**:
   - The degree of a polynomial is the highest exponent of the variable \( x \).
   - In this case, the highest exponent is \( 6 \), so the degree of the polynomial is **6**.

2. **Leading Coefficient**:
   - The leading coefficient is the coefficient of the term with the highest degree.
   - The term with the highest degree is \( 5x^6 \), and the coefficient is **5**.

3. **Maximum Number of Real Zeros**:
   - The maximum number of real zeros of a polynomial is at most equal to its degree.
   - Since the degree is \( 6 \), the maximum number of real zeros is **6**.

### Summary:
- **Degree**: 6
- **Leading Coefficient**: 5
- **Maximum number of real zeros**: 6

Would you like any additional details or explanations on this problem?

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Here are some related questions:
1. How do we determine if all the zeros are real for a polynomial?
2. Can a polynomial of degree 6 have fewer than 6 real zeros? Why or why not?
3. What is the effect of a leading coefficient on the end behavior of a polynomial?
4. How do you find the actual real zeros of a polynomial?
5. What happens to the number of real zeros if the polynomial is factored?

**Tip**: The Fundamental Theorem of Algebra tells us that the number of complex roots (including real ones) of a polynomial is always equal to its degree.

Find the degree, leading coefficients, and the maximum number of real zeros of the polynomial: f(x) = 5x^6 + 2x^5 + 1 - 4x^3

Learn how to find the degree, leading coefficient, and the maximum number of real zeros of the polynomial f(x) = 5x^6 + 2x^5 + 1 - 4x^3. This problem covers polynomial concepts, including identifying degree and leading coefficient. Suitable for students in Grades 9-12.

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