Answer the questions about the following polynomial.

one half, plus, 5, x, to the power 4 , minus, 7, x, plus, x, squared

21​+5*x*4−7*x*+*x*2

Answer

Attempt 1 out of 2

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

Let’s analyze the polynomial expression:  
\[
\frac{1}{2} + 5x^4 - 7x + x^2
\]

1. **Degree of the Polynomial**:
   - The term with the highest power of \( x \) is \( 5x^4 \).
   - Therefore, the polynomial is **quartic**, as the highest exponent is 4.

2. **Number of Terms**:
   - The polynomial consists of four terms: \( \frac{1}{2} \), \( 5x^4 \), \( -7x \), and \( x^2 \).

3. **Constant Term**:
   - The constant term (the term without any \( x \)) is \( \frac{1}{2} \).

4. **Leading Term**:
   - The term with the highest power of \( x \) is \( 5x^4 \), which is the **leading term**.

5. **Leading Coefficient**:
   - The leading coefficient (the coefficient of the leading term) is \( 5 \).

### Summary Answer:
- **Degree**: Quartic
- **Number of Terms**: 4
- **Constant Term**: \( \frac{1}{2} \)
- **Leading Term**: \( 5x^4 \)
- **Leading Coefficient**: 5

Would you like further explanation on any part of this answer?

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Here are five related questions for further practice:
1. What is the degree of a polynomial with a leading term of \( 7x^5 \)?
2. How do you determine the leading term and leading coefficient in any polynomial?
3. What is the difference between a constant term and a leading term?
4. For a polynomial \( 2x^3 - 4x + 5 \), identify its leading term and constant term.
5. Why is the term "quadratic" used for polynomials with degree 2?

**Tip**: Remember, the degree of a polynomial is determined by its term with the highest power of \( x \).

Answer the questions about the following polynomial: one half, plus, 5, x, to the power 4, minus, 7, x, plus, x, squared

Explore the characteristics of the polynomial 1/2 + 5x^4 - 7x + x^2, including its degree, number of terms, constant term, leading term, and leading coefficient. This analysis is suitable for students in Grades 8-10.

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