Answer the questions about the following polynomial.

minus, start fraction, x, to the power 4 , divided by, 5, end fraction, minus, 1

−5*x*4​−1

Answer

Attempt 1 out of 2

The expression represents a  linearcubicquadraticquinticquartic polynomial with  terms. The constant term is , the leading term is , and the leading coefficient is .

Let's analyze the given polynomial expression:

\[
-\frac{x^4}{5} - 1
\]

### Analyzing the Polynomial

1. **Degree of the Polynomial**: The degree of a polynomial is the highest power of the variable \(x\) in the expression. In this case, the highest power of \(x\) is 4 (from the term \(-\frac{x^4}{5}\)). Therefore, the polynomial is of degree **4**. This means it is a **quartic** polynomial.

2. **Number of Terms**: A polynomial is made up of terms separated by addition or subtraction. The given polynomial has two terms:
   - \(-\frac{x^4}{5}\)
   - \(-1\)

   So, there are **2 terms**.

3. **Constant Term**: The constant term in a polynomial is the term without any variable (without \(x\)). In this polynomial, the constant term is **\(-1\)**.

4. **Leading Term**: The leading term of a polynomial is the term with the highest degree. Here, the leading term is **\(-\frac{x^4}{5}\)**.

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

### Filling in the Blanks
Based on the analysis above:
- The expression represents a **quartic** polynomial with **2** terms.
- The constant term is **\(-1\)**.
- The leading term is **\(-\frac{x^4}{5}\)**.
- The leading coefficient is **\(-\frac{1}{5}\)**.

Would you like more details or have any other questions?

### Related Questions to Expand Understanding:
1. How do you classify polynomials based on their degree?
2. What is the difference between a quartic and a cubic polynomial?
3. How do you determine the leading coefficient in a polynomial?
4. What are some common methods to solve quartic polynomials?
5. How does the sign of the leading coefficient affect the graph of a polynomial?

**Tip:** When dealing with fractions in polynomials, always identify the coefficient separately from the variable and its exponent for clarity.

Understand and analyze the quartic polynomial expression -x^4/5 - 1. Learn about its degree, number of terms, constant term, leading term, and leading coefficient with detailed explanations.

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