Let's analyze the given polynomial step-by-step to answer the questions.
The polynomial given is:
[tex]\[
\frac{1}{7} x^5 - 3 + 9x
\][/tex]
1. Type of Polynomial:
The given polynomial consists of terms with non-negative integer exponents. Since the highest exponent of the variable [tex]\( x \)[/tex] is 5, it is a polynomial of degree 5. Specifically, it is a quintic polynomial.
2. Number of Terms:
By breaking down the polynomial, we can identify the individual terms:
[tex]\[
\frac{1}{7} x^5, \quad 9x, \quad -3
\][/tex]
Hence, there are 3 terms in this polynomial.
3. Constant Term:
The constant term in a polynomial is the term without the variable [tex]\( x \)[/tex]. In this polynomial, the term without the variable [tex]\( x \)[/tex] is [tex]\(-3\)[/tex]. Thus, the constant term is:
[tex]\[
-3
\][/tex]
4. Leading Term:
The leading term of a polynomial is the term with the highest power of [tex]\( x \)[/tex]. In this polynomial, the term with the highest power of [tex]\( x \)[/tex] is:
[tex]\[
\frac{1}{7} x^5
\][/tex]
5. Leading Coefficient:
The leading coefficient is the coefficient of the leading term, which is the term with the highest power of [tex]\( x \)[/tex]. In this case, the leading term is [tex]\(\frac{1}{7} x^5\)[/tex], so the leading coefficient is:
[tex]\[
\frac{1}{7}
\][/tex]
When converted to a decimal, this is approximately [tex]\(0.142857142857143\)[/tex].
Summarizing the detailed analysis:
- The expression represents a quintic polynomial.
- The polynomial has 3 terms.
- The constant term is [tex]\(-3\)[/tex].
- The leading term is [tex]\(0.142857142857143 x^5\)[/tex].
- The leading coefficient is [tex]\(0.142857142857143\)[/tex].
