The problem is asking us to analyze the function $f(x) = 2x + x^5$ to determine if it is a polynomial function. If it is, we should state its degree, identify the leading term, the constant term, and write it in standard form.

Solution

1. Determine if it’s a polynomial function:
   A polynomial function is an expression that consists of terms in the form $ax^n$, where $a$ is a coefficient and $n$ is a non-negative integer. The function $f(x) = 2x + x^5$ meets these criteria since it has terms $2x$ and $x^5$ (both with non-negative integer exponents).

2. Standard Form:
   The standard form of a polynomial arranges terms in descending order of their powers. Thus, we rewrite $f(x) = x^5 + 2x$.

3. Degree of the Polynomial:
   The degree of a polynomial is the highest power of $x$. In this case, the highest power is $5$, so the degree of this polynomial is $5$.

4. Leading Term:
   The leading term is the term with the highest degree, which is $x^5$.

5. Constant Term:
   A constant term is a term with no variable attached. Since there is no constant term in $f(x) = x^5 + 2x$, the constant term is $0$.

Summary

• Polynomial Function: Yes
• Standard Form: $f(x) = x^5 + 2x$
• Degree: 5
• Leading Term: $x^5$
• Constant Term: $0$

Would you like further details or have any questions?

Here are some related questions to explore:

1. How do we determine if a function is a polynomial?
2. What are examples of non-polynomial functions?
3. How do we find the degree of a polynomial with multiple terms?
4. Why is the highest power term called the "leading term"?
5. How does the constant term affect the polynomial function graph?

Tip: When writing polynomials in standard form, always order terms by descending powers of $x$ for clarity.