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.