Sure! Let's break down the given polynomial function [tex]\( P(x) = 11 - x - 2x^3 + 9x^5 \)[/tex] and find the required elements.
### 1. Degree of the Function
The degree of a polynomial is determined by the highest power of [tex]\( x \)[/tex] in the polynomial.
In the polynomial [tex]\( P(x) = 11 - x - 2x^3 + 9x^5 \)[/tex], the highest power of [tex]\( x \)[/tex] is 5. Therefore, the degree of the function is:
[tex]\[ \boxed{5} \][/tex]
### 2. Number of Terms
The number of terms in a polynomial is the count of distinct monomials (individual terms) in the polynomial expression.
In [tex]\( P(x) = 11 - x - 2x^3 + 9x^5 \)[/tex], there are 4 distinct terms:
1. [tex]\( 11 \)[/tex]
2. [tex]\( -x \)[/tex]
3. [tex]\( -2x^3 \)[/tex]
4. [tex]\( 9x^5 \)[/tex]
Therefore, the number of terms is:
[tex]\[ \boxed{4} \][/tex]
### 3. Leading Coefficient
The leading coefficient of a polynomial is the coefficient of the term with the highest power of [tex]\( x \)[/tex].
In [tex]\( P(x) = 11 - x - 2x^3 + 9x^5 \)[/tex], the term with the highest power of [tex]\( x \)[/tex] is [tex]\( 9x^5 \)[/tex], where the coefficient is 9. Therefore, the leading coefficient is:
[tex]\[ \boxed{9} \][/tex]
### 4. Constant Term
The constant term in a polynomial is the term that does not contain any [tex]\( x \)[/tex].
In [tex]\( P(x) = 11 - x - 2x^3 + 9x^5 \)[/tex], the constant term is [tex]\( 11 \)[/tex]. Therefore, the constant term is:
[tex]\[ \boxed{11} \][/tex]
Thus, summarizing the answers:
- Degree of the function: [tex]\(\boxed{5}\)[/tex]
- Number of terms: [tex]\(\boxed{4}\)[/tex]
- Leading coefficient: [tex]\(\boxed{9}\)[/tex]
- Constant term: [tex]\(\boxed{11}\)[/tex]
