https://file.aiquickdraw.com/mathbot/file/17292151601542nnl7007.jpg

Let's work through the polynomial provided in the image, which is:

\[
2x^3 + 2x - 4x^4 - x^2 + 8
\]

### Identifying the coefficients and degree of each term:

1. **First term: \(2x^3\)**
   - Degree: The exponent of \(x\) is \(3\), so the degree is **3**.
   - Coefficient: The coefficient of this term is **2**.

2. **Second term: \(2x\)**
   - Degree: The exponent of \(x\) is \(1\), so the degree is **1**.
   - Coefficient: The coefficient is **2**.

3. **Third term: \(-4x^4\)**
   - Degree: The exponent of \(x\) is \(4\), so the degree is **4**.
   - Coefficient: The coefficient is **-4**.

4. **Fourth term: \(-x^2\)**
   - Degree: The exponent of \(x\) is \(2\), so the degree is **2**.
   - Coefficient: The coefficient is **-1**.

5. **Fifth term: \(8\)**
   - Degree: This is a constant term, so the degree is **0**.
   - Coefficient: The coefficient is **8**.

### Leading term and leading coefficient:

- The **leading term** is the term with the highest degree, which is \(-4x^4\).
- The **leading coefficient** is the coefficient of the leading term, which is **-4**.

### Degree of the polynomial:

- The **degree of the polynomial** is the degree of the leading term, which is **4**.

---

Here’s a summary of the answers for the table:

1. **First term:**
   - Degree: 3
   - Coefficient: 2
2. **Second term:**
   - Degree: 1
   - Coefficient: 2
3. **Third term:**
   - Degree: 4
   - Coefficient: -4
4. **Fourth term:**
   - Degree: 2
   - Coefficient: -1
5. **Fifth term:**
   - Degree: 0
   - Coefficient: 8

**Leading Coefficient**: -4  
**Degree of the Leading Term**: 4  
**Degree of the Polynomial**: 4

---

Would you like more details on any of these calculations or have any questions?

Here are some related questions you could explore:
1. What is the difference between a polynomial’s degree and the degree of its terms?
2. How would the polynomial change if the highest degree term were missing?
3. How does the leading coefficient affect the graph of a polynomial?
4. What is the significance of the constant term in a polynomial?
5. How do you determine if a polynomial is classified as monomial, binomial, or trinomial?

**Tip:** Always arrange polynomials in descending order of degrees to easily identify the leading term.

Given the polynomial, identify the coefficients and degree of each term: 2x³ + 2x – 4x⁴ – x² + 8.

Analyze the polynomial 2x³ + 2x – 4x⁴ – x² + 8 by identifying the degree and coefficients of each term. Learn how to determine the leading term and the overall degree of a polynomial. Ideal for students in Grades 8-10.

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