Answer the questions about the following polynomial.

minus, x, cubed, minus, 10, x, plus, one half

−*x*3−10*x*+21​

Answer

Attempt 1 out of 2

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

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.

Learn how to analyze the polynomial expression -x^3 - 10x + 1/2. Understand its degree, number of terms, constant term, leading term, and leading coefficient. Suitable for high school students studying polynomial functions and algebraic expressions.

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