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 given function: \( f(x) = 4x^3 + \frac{1}{2}x^{-1} - 2x + 1 \).
2. Recall 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.
3. Examine each term in the function: \( 4x^3 \), \( \frac{1}{2}x^{-1} \), \( -2x \), and \( 1 \).
4. Notice that the term \( \frac{1}{2}x^{-1} \) involves a negative exponent, which violates the condition for a polynomial function.
5. Conclude that since the function contains a term with a negative exponent, it is not a polynomial function.