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. Identify the terms in the function f(x) = 4x^3 + \frac{1}{2}x^{-1} - 2x + 1.
2. Recall that a polynomial function is a sum of terms of the form a_n x^n, where n is a non-negative integer and a_n is a coefficient.
3. Examine each term: 4x^3 is a polynomial term with n=3, -2x is a polynomial term with n=1, and 1 is a constant term with n=0.
4. Notice the term \frac{1}{2}x^{-1}. The exponent -1 is negative, which means this term is not a polynomial term.
5. Conclude that because the function includes a term with a negative exponent, it is not a polynomial function.