Question
3. \( f(x)=-2 x^{3}-3 x+4 x^{5}-10 \) SF: Number of Terms: Degree: Leading Coefficient:
Ask by Craig Byrd. in the United States
Feb 27,2025
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Tutor-Verified Answer
Answer
- Number of Terms: 4
- Degree: 5
- Leading Coefficient: 4
Solution
To analyze the polynomial function \( f(x) = -2x^3 - 3x + 4x^5 - 10 \), we will extract the following information:
1. **Number of Terms**: Count the distinct terms in the polynomial.
2. **Degree**: Identify the highest power of \( x \) in the polynomial.
3. **Leading Coefficient**: Determine the coefficient of the term with the highest degree.
Let's break it down step by step:
### Step 1: Number of Terms
The polynomial can be expressed as:
- \( -2x^3 \)
- \( -3x \)
- \( 4x^5 \)
- \( -10 \)
Counting these, we have 4 distinct terms.
### Step 2: Degree
The degree of a polynomial is the highest exponent of \( x \). In this polynomial, the highest exponent is 5 (from the term \( 4x^5 \)).
### Step 3: Leading Coefficient
The leading coefficient is the coefficient of the term with the highest degree. Here, the leading term is \( 4x^5 \), so the leading coefficient is 4.
### Summary
- **Number of Terms**: 4
- **Degree**: 5
- **Leading Coefficient**: 4
Thus, the final answers are:
- Number of Terms: 4
- Degree: 5
- Leading Coefficient: 4
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Simplify this solution The Deep Dive
Let's break it down! The given polynomial \( f(x) = -2x^3 - 3x + 4x^5 - 10 \) is a fun mix of terms. It has **four terms** in total. The highest power of \( x \) present here is \( 5 \), so the **degree** of the polynomial is \( 5 \). Now, looking at the leading term, we see that \( 4x^5 \) is the star of the show, giving us a **leading coefficient** of \( 4 \). This polynomial can be graphed to visualize its behavior! You might notice that it has a mix of rising and falling sections, all thanks to the varying powers of \( x \). The leading coefficient being positive means, as \( x \) becomes very large, \( f(x) \) will head towards infinity!
