To analyze the polynomial [tex]\( 7x^4 - x^3 + 2x + \frac{1}{4} \)[/tex], let's break it down step-by-step.
1. Identify the Degree and Type of Polynomial:
- The given polynomial is [tex]\( 7x^4 - x^3 + 2x + \frac{1}{4} \)[/tex].
- The degree of the polynomial is determined by the highest power of [tex]\( x \)[/tex], which is [tex]\( 4 \)[/tex]. Therefore, it is a polynomial of degree [tex]\( 4 \)[/tex].
2. Determine the Number of Terms:
- The polynomial consists of the terms: [tex]\( 7x^4 \)[/tex], [tex]\( -x^3 \)[/tex], [tex]\( 2x \)[/tex], and [tex]\( \frac{1}{4} \)[/tex].
- Counting these, we find the polynomial has [tex]\( 4 \)[/tex] terms.
3. Identify the Constant Term:
- The constant term is the term that does not contain the variable [tex]\( x \)[/tex].
- In this polynomial, the constant term is [tex]\( \frac{1}{4} \)[/tex].
4. Identify the Leading Term:
- The leading term is the term containing the highest power of [tex]\( x \)[/tex].
- Here, the leading term is [tex]\( 7x^4 \)[/tex].
5. Identify the Leading Coefficient:
- The leading coefficient is the coefficient of the leading term.
- For the term [tex]\( 7x^4 \)[/tex], the leading coefficient is [tex]\( 7 \)[/tex].
Therefore, summarizing the information:
- The polynomial is a fourth-degree polynomial with 4 terms.
- The constant term is [tex]\( \frac{1}{4} \)[/tex].
- The leading term is [tex]\( 7x^4 \)[/tex].
- The leading coefficient is [tex]\( 7 \)[/tex].
Now, completing the provided template:
The expression represents a fourth-degree polynomial with 4 terms. The constant term is [tex]\(\frac{1}{4}\)[/tex], the leading term is [tex]\( 7x^4 \)[/tex], and the leading coefficient is 7.
