Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

• A. Polynomial with $n=3,a_{n}=4$
• B. Polynomial with $n=4,a_{n}=3$
• C. Polynomial with $n=-1,a_{n}=\frac12$
• D. Not a polynomial function.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the definition of a polynomial function. A polynomial function is an expression consisting of variables and coefficients, involving only non-negative integer powers of the variable.
2. Step 2: Examine the given function f(x) = 4x^3 + \frac{1}{2}x^{-1} - 2x + 1. Identify the terms and their respective powers.
3. Step 3: Check each term to see if it meets the criteria for a polynomial. The term \frac{1}{2}x^{-1} involves a negative exponent, which is not allowed in a polynomial function.
4. Step 4: Since the function contains a term with a negative exponent, it does not meet the criteria for a polynomial function.
5. Step 5: Conclude that the given function is not a polynomial function because it includes a term with a negative exponent.