To analyze the given polynomial function [tex]\( f(x) = \frac{\pi}{3} x^6 + 6 - 5 x^4 \)[/tex], we will determine the degree, leading term, and leading coefficient.
First, let's write the polynomial in standard form:
[tex]\[ f(x) = \frac{\pi}{3} x^6 + 6 - 5 x^4 \][/tex]
### Step 1: Determining the Degree of the Polynomial
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] present in the expression.
In the given polynomial:
- The term [tex]\(\frac{\pi}{3} x^6\)[/tex] has the exponent 6.
- The term [tex]\(6\)[/tex] is a constant and has an implicit exponent of 0 (i.e., [tex]\(6 = 6x^0\)[/tex]).
- The term [tex]\(-5 x^4\)[/tex] has the exponent 4.
Among these terms, the highest exponent is 6.
Therefore, the degree of the polynomial is [tex]\( \boxed{6} \)[/tex].
### Step 2: Identifying the Leading Term
The leading term of a polynomial is the term with the highest exponent.
From our findings, the term with the highest exponent [tex]\( 6 \)[/tex] is [tex]\(\frac{\pi}{3} x^6\)[/tex].
Therefore, the leading term is [tex]\( \boxed{\frac{\pi}{3} x^6} \)[/tex].
### Step 3: Finding the Leading Coefficient
The leading coefficient is the coefficient of the leading term.
In the leading term [tex]\(\frac{\pi}{3} x^6\)[/tex], the coefficient is [tex]\(\frac{\pi}{3}\)[/tex].
Therefore, the leading coefficient is [tex]\( \boxed{\frac{\pi}{3}} \)[/tex].
Revisiting the numerical values based on the analysis:
- The degree of the polynomial is [tex]\(6\)[/tex].
- The leading term is [tex]\( \frac{\pi}{3} x^6 \)[/tex], which provides a numerical value of approximately [tex]\(1.0471975511966 x^6\)[/tex].
- The leading coefficient is [tex]\(\frac{\pi}{3}\)[/tex], approximately [tex]\(1.04719755119660\)[/tex].
So, summarizing the final results:
- Degree: [tex]\( \boxed{6} \)[/tex]
- Leading Term: [tex]\( \boxed{1.0471975511966 x^6} \)[/tex]
- Leading Coefficient: [tex]\( \boxed{1.04719755119660} \)[/tex]
