Let's analyze the given polynomial expression:

$-x^3 - 10x + \frac{1}{2}$

Analyzing the Polynomial

1. Degree of the Polynomial: The degree of a polynomial is determined by the highest power of the variable $x$ in the expression. In this case, the highest power of $x$ is 3 (from the term $-x^3$), so the polynomial is of degree 3. This means it is a cubic polynomial.

2. Number of Terms: A polynomial consists of terms separated by addition or subtraction. The given polynomial has three terms:

   • $-x^3$
   • $-10x$
   • $\frac{1}{2}$

   So, there are 3 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 $\frac{1}{2}$.

4. Leading Term: The leading term of a polynomial is the term with the highest degree. In this case, the leading term is $-x^3$.

5. Leading Coefficient: The leading coefficient is the coefficient of the leading term. The leading term is $-x^3$, so the leading coefficient is -1.

Filling in the Blanks

Based on the analysis above:

• The expression represents a cubic polynomial with 3 terms.
• The constant term is $\frac{1}{2}$.
• The leading term is $-x^3$.
• The leading coefficient is -1.

Would you like more details or have any other questions?

Related Questions to Expand Understanding:

1. What is the degree of a polynomial and how is it determined?
2. How can you identify the leading term and coefficient in any polynomial?
3. What are the characteristics of a cubic polynomial in terms of its graph and roots?
4. How do you determine the constant term of a polynomial?
5. What are the possible number of roots for a cubic polynomial?

Tip: Always arrange polynomials in descending order of their degrees to easily identify their characteristics, such as the leading term and coefficient.