Determine the leading​ term, the leading​ coefficient, and the degree of the polynomial. Then classify the polynomial as​ constant, linear,​ quadratic, cubic, or quartic.
g\left(x\right)=-\frac{1}{5}x^{4}-5x+8

Let's analyze the given polynomial:  
\[
g(x) = -\frac{1}{5}x^4 - 5x + 8
\]

### Step 1: Determine the Leading Term
The **leading term** is the term with the highest power of \(x\).  
- In this polynomial, the term with the highest power is \(-\frac{1}{5}x^4\).

### Step 2: Determine the Leading Coefficient
The **leading coefficient** is the coefficient of the leading term.  
- The leading term is \(-\frac{1}{5}x^4\), so the leading coefficient is \(-\frac{1}{5}\).

### Step 3: Determine the Degree of the Polynomial
The **degree** of a polynomial is the highest exponent of the variable \(x\).  
- In this case, the highest exponent is \(4\), so the degree is \(4\).

### Step 4: Classify the Polynomial
Based on its degree, we can classify the polynomial:
- A polynomial of degree 0 is **constant**.
- A polynomial of degree 1 is **linear**.
- A polynomial of degree 2 is **quadratic**.
- A polynomial of degree 3 is **cubic**.
- A polynomial of degree 4 is **quartic**.

Since this polynomial has a degree of 4, it is classified as **quartic**.

### Summary
- **Leading Term:** \(-\frac{1}{5}x^4\)
- **Leading Coefficient:** \(-\frac{1}{5}\)
- **Degree:** 4
- **Classification:** Quartic

Would you like more details or have any questions?

### Related Questions:
1. How do you determine the leading term of a polynomial with multiple variables?
2. What happens to the leading term if the polynomial is rearranged?
3. Can a polynomial have a negative degree?
4. How does the degree of a polynomial affect its end behavior?
5. What is the leading term of the polynomial \(3x^3 - 7x + 2\)?

### Tip:
Always look for the term with the highest exponent when determining the degree and leading term of a polynomial. This helps in analyzing the polynomial's behavior.

Determine the leading​ term, the leading​ coefficient, and the degree of the polynomial. Then classify the polynomial as​ constant, linear,​ quadratic, cubic, or quartic.
g(x) = -\frac{1}{5}x^4 - 5x + 8

Analyze the polynomial g(x) = -1/5x^4 - 5x + 8 by identifying its leading term, leading coefficient, and degree. This detailed solution explains the steps to classify the polynomial as constant, linear, quadratic, cubic, or quartic. Ideal for high school students in Grades 9-12.

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