A polynomial has the end behavior lim
𝑥→−∞ 𝑦 = −∞ and lim
𝑥→+∞ 𝑦 = −∞. It has four x-
intercepts, one absolute maximum, two other relative maxima, and two relative minima.
Describe everything you know about the leading coefficient, degree, and real and non-
real zeroes of the function.

Given the polynomial's end behavior and other characteristics, we can deduce several properties about the polynomial's leading coefficient, degree, and its real and non-real zeroes. Let's break this down:

### 1. **End Behavior**
- **Given**: 
  - \(\lim_{x \to -\infty} y = -\infty\)
  - \(\lim_{x \to +\infty} y = -\infty\)

- **Implication**: The end behavior tells us that as \(x\) approaches both negative and positive infinity, the function approaches negative infinity. This is characteristic of a polynomial of **even degree** with a **negative leading coefficient**. For example, a quadratic polynomial like \(y = -x^2\) has this end behavior.

### 2. **Degree of the Polynomial**
- **Given**:
  - The polynomial has **four \(x\)-intercepts**.
  
- **Implication**: 
  - Since the polynomial has four \(x\)-intercepts, the degree must be at least 4. Each \(x\)-intercept corresponds to a real root, and the minimum degree of a polynomial with 4 real roots is 4.
  - Considering the given end behavior, the degree must be even. So, the degree is at least 4, but it could be higher (e.g., 6, 8) depending on whether some roots are complex or if there's any multiplicity among the roots.

### 3. **Leading Coefficient**
- **Given**: 
  - The end behavior is that both as \(x \to -\infty\) and \(x \to +\infty\), \(y \to -\infty\).
  
- **Implication**: 
  - This indicates that the leading coefficient is **negative**. A negative leading coefficient causes the polynomial to drop off to negative infinity on both ends if the degree is even.

### 4. **Number and Nature of Zeroes**
- **Given**:
  - **Four \(x\)-intercepts**: This means there are four real roots.
  - **One absolute maximum**: The highest point on the graph.
  - **Two other relative maxima**: These are peaks, but not the highest point.
  - **Two relative minima**: These are valleys, lower points on the graph.

- **Implication**:
  - The polynomial has **four real roots** (corresponding to the \(x\)-intercepts).
  - The total number of turning points (maxima and minima combined) is 5, which is consistent with a polynomial of **degree 6** (since a degree \(n\) polynomial can have at most \(n-1\) turning points).
  - Since the degree is even and the function has the described end behavior, the degree is most likely 6.
  - Given the four real roots, and knowing the degree is 6, this implies there are two **non-real (complex) roots**. Complex roots in polynomials with real coefficients come in conjugate pairs, so the two non-real roots must be conjugates.

### **Summary**
- **Leading Coefficient**: Negative.
- **Degree**: Most likely 6 (even).
- **Real Zeroes**: Four real roots corresponding to the four \(x\)-intercepts.
- **Non-Real Zeroes**: Two complex roots, which must be a conjugate pair.

Would you like more details on any part of this analysis or have any other questions?

Here are 5 related questions for further exploration:
1. How does the leading coefficient of a polynomial affect its end behavior?
2. What is the maximum number of turning points a polynomial of degree 7 can have?
3. Can a polynomial have only complex roots? Explain the conditions.
4. How can you determine the degree of a polynomial by looking at its graph?
5. What are the differences between absolute maxima and relative maxima in a function?

**Tip:** The degree of a polynomial not only tells us about the number of roots (real and complex) but also the number of possible turning points, which can give insight into the graph's shape.

Explore the characteristics of a polynomial with specified end behavior, x-intercepts, and extrema. Learn about the degree, leading coefficient, real and non-real zeroes, and their implications. Suitable for high school students studying polynomial functions.

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