Certainly! Let's analyze the polynomial [tex]\( 1 + \frac{1}{3} x^4 \)[/tex].
1. Identifying the Type of Polynomial:
The polynomial [tex]\( 1 + \frac{1}{3} x^4 \)[/tex] contains terms with powers of [tex]\( x \)[/tex] (specifically, [tex]\( x^4 \)[/tex] and the constant term). Since the highest power of [tex]\( x \)[/tex] here is [tex]\( 4 \)[/tex], it classifies this polynomial as a general polynomial rather than a monomial.
2. Counting the Number of Terms:
The polynomial [tex]\( 1 + \frac{1}{3} x^4 \)[/tex] consists of two distinct terms: the constant term [tex]\( 1 \)[/tex] and the term involving [tex]\( x \)[/tex], which is [tex]\( \frac{1}{3} x^4 \)[/tex]. Therefore, this polynomial has 2 terms.
3. Identifying the Constant Term:
The constant term in a polynomial is the term without any variable attached to it. Here, the constant term is [tex]\( 1 \)[/tex].
4. Identifying the Leading Term:
The leading term in a polynomial is the term with the highest power of [tex]\( x \)[/tex]. In this polynomial, the leading term is [tex]\( \frac{1}{3} x^4 \)[/tex].
5. Identifying the Leading Coefficient:
The leading coefficient is the coefficient of the leading term. For the leading term [tex]\( \frac{1}{3} x^4 \)[/tex], the coefficient is [tex]\( \frac{1}{3} \)[/tex].
Summarizing the answers to the questions:
- The expression represents a general polynomial with 2 terms.
- The constant term is 1 , the leading term is [tex]\(\frac{1}{3} x^4\)[/tex] , and the leading coefficient is [tex]\(\frac{1}{3}\)[/tex].
