Answer :
Certainly! Let's break this down step-by-step.
The polynomial given is:
[tex]\[ 3x^9 + 7x^7 - 3x^3 + 2 \][/tex]
### 1. Identifying the Coefficients and Degrees of Each Term
- Term: [tex]\( 3x^9 \)[/tex]
- Coefficient: 3
- Degree: 9
- Term: [tex]\( 7x^7 \)[/tex]
- Coefficient: 7
- Degree: 7
- Term: [tex]\( -3x^3 \)[/tex]
- Coefficient: -3
- Degree: 3
- Term: [tex]\( 2 \)[/tex]
- Coefficient: 2
- Degree: 0
### 2. Summarizing the Coefficients and Degrees
Let's summarize the information we gathered.
- Coeficient of [tex]\( 3x^9 \)[/tex]: 3
- Degree of [tex]\( 3x^9 \)[/tex]: 9
- Coefficient of [tex]\( 7x^7 \)[/tex]: 7
- Degree of [tex]\( 7x^7 \)[/tex]: 7
- Coefficient of [tex]\( -3x^3 \)[/tex]: -3
- Degree of [tex]\( -3x^3 \)[/tex]: 3
- Coefficient of [tex]\( 2 \)[/tex]: 2
- Degree of [tex]\( 2 \)[/tex]: 0
### 3. Determining the Degree of the Polynomial
The degree of a polynomial is the highest degree of its terms. The degrees of the terms in this polynomial are 9, 7, 3, and 0. Therefore, the highest degree is:
[tex]\[ \text{Degree of the polynomial} = 9 \][/tex]
### 4. Finding the Leading Term
The leading term of a polynomial is the term with the highest degree. Here, the term with the highest degree (9) is [tex]\( 3x^9 \)[/tex].
Therefore, the leading term of the polynomial is:
[tex]\[ 3x^9 \][/tex]
### Summary:
- The coefficient of the term [tex]\( 3x^9 \)[/tex] is 3.
- The degree of the term [tex]\( 3x^9 \)[/tex] is 9.
- The coefficient of the term [tex]\( 7x^7 \)[/tex] is 7.
- The degree of the term [tex]\( 7x^7 \)[/tex] is 7.
- The coefficient of the term [tex]\( -3x^3 \)[/tex] is -3.
- The degree of the term [tex]\( -3x^3 \)[/tex] is 3.
- The coefficient of the term [tex]\( 2 \)[/tex] is 2.
- The degree of the term [tex]\( 2 \)[/tex] is 0.
- The degree of the polynomial [tex]\( 3x^9 + 7x^7 - 3x^3 + 2 \)[/tex] is 9.
- The leading term of the polynomial [tex]\( 3x^9 + 7x^7 - 3x^3 + 2 \)[/tex] is 3x^9.
This concludes the detailed breakdown of the problem! If you have any more questions or need further clarification, feel free to ask!
The polynomial given is:
[tex]\[ 3x^9 + 7x^7 - 3x^3 + 2 \][/tex]
### 1. Identifying the Coefficients and Degrees of Each Term
- Term: [tex]\( 3x^9 \)[/tex]
- Coefficient: 3
- Degree: 9
- Term: [tex]\( 7x^7 \)[/tex]
- Coefficient: 7
- Degree: 7
- Term: [tex]\( -3x^3 \)[/tex]
- Coefficient: -3
- Degree: 3
- Term: [tex]\( 2 \)[/tex]
- Coefficient: 2
- Degree: 0
### 2. Summarizing the Coefficients and Degrees
Let's summarize the information we gathered.
- Coeficient of [tex]\( 3x^9 \)[/tex]: 3
- Degree of [tex]\( 3x^9 \)[/tex]: 9
- Coefficient of [tex]\( 7x^7 \)[/tex]: 7
- Degree of [tex]\( 7x^7 \)[/tex]: 7
- Coefficient of [tex]\( -3x^3 \)[/tex]: -3
- Degree of [tex]\( -3x^3 \)[/tex]: 3
- Coefficient of [tex]\( 2 \)[/tex]: 2
- Degree of [tex]\( 2 \)[/tex]: 0
### 3. Determining the Degree of the Polynomial
The degree of a polynomial is the highest degree of its terms. The degrees of the terms in this polynomial are 9, 7, 3, and 0. Therefore, the highest degree is:
[tex]\[ \text{Degree of the polynomial} = 9 \][/tex]
### 4. Finding the Leading Term
The leading term of a polynomial is the term with the highest degree. Here, the term with the highest degree (9) is [tex]\( 3x^9 \)[/tex].
Therefore, the leading term of the polynomial is:
[tex]\[ 3x^9 \][/tex]
### Summary:
- The coefficient of the term [tex]\( 3x^9 \)[/tex] is 3.
- The degree of the term [tex]\( 3x^9 \)[/tex] is 9.
- The coefficient of the term [tex]\( 7x^7 \)[/tex] is 7.
- The degree of the term [tex]\( 7x^7 \)[/tex] is 7.
- The coefficient of the term [tex]\( -3x^3 \)[/tex] is -3.
- The degree of the term [tex]\( -3x^3 \)[/tex] is 3.
- The coefficient of the term [tex]\( 2 \)[/tex] is 2.
- The degree of the term [tex]\( 2 \)[/tex] is 0.
- The degree of the polynomial [tex]\( 3x^9 + 7x^7 - 3x^3 + 2 \)[/tex] is 9.
- The leading term of the polynomial [tex]\( 3x^9 + 7x^7 - 3x^3 + 2 \)[/tex] is 3x^9.
This concludes the detailed breakdown of the problem! If you have any more questions or need further clarification, feel free to ask!
