To determine if the given expression [tex]\(-p^5 \sqrt{7}\)[/tex] is a polynomial, we should analyze its components and structure carefully.
### Step-by-Step Solution:
1. Identify the Components:
- The given expression is [tex]\(-p^5 \sqrt{7}\)[/tex].
2. Break Down the Expression:
- [tex]\(-p^5\)[/tex]: This part of the expression involves a variable [tex]\(p\)[/tex] raised to the power of 5, with a coefficient of -1.
- [tex]\(\sqrt{7}\)[/tex]: This is a constant term because it does not involve any variable; it's just a numerical value.
3. Combine the Components:
- When we multiply the term [tex]\(-p^5\)[/tex] by the constant [tex]\(\sqrt{7}\)[/tex], we get [tex]\(-p^5 \sqrt{7}\)[/tex].
4. Definition of a Polynomial:
- A polynomial is an expression composed of variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
- Importantly, the exponents must be non-negative integers, and each term must involve only these kinds of operations.
5. Determine if Each Component Meets the Criteria:
- [tex]\(-p^5\)[/tex] meets the criteria since [tex]\(p\)[/tex] is raised to the 5th power, which is a non-negative integer, and it is multiplied by -1.
- Multiplying by [tex]\(\sqrt{7}\)[/tex], a constant, does not change the fact that the term [tex]\(-p^5\)[/tex] still meets the polynomial criteria. Constants are permissible in polynomials.
### Conclusion:
Based on the analysis, the expression [tex]\(-p^5 \sqrt{7}\)[/tex] is indeed a polynomial because it is composed of a single term where the variable [tex]\(p\)[/tex] is raised to the non-negative integer exponent 5, and it is multiplied by a constant [tex]\(\sqrt{7}\)[/tex].
- Type of Polynomial: Since there is only one term in the expression, it is called a monomial.
- Degree of Polynomial: The degree of a polynomial is determined by the highest power of the variable present in the expression. In this case, the highest power of [tex]\(p\)[/tex] is 5.
### Final Answer:
The given expression is a polynomial. It is a monomial with a degree of 5.
