Question

decide whether $p(x) = \frac{1}{2}x^2 + 3x - 4x^3 + 6x^4 - 1$ is a polynomial function.
\bigcirc polynomial function
\bigcirc not a polynomial function
if so, write the polynomial function in standard form; otherwise leave the box empty. write any fractional coefficients in simplest form.
$p(x) = \square$
state the degree, type, and leading coefficient of the polynomial function. if the function is not a polynomial function, leave the boxes blank.
degree: \bigcirc
type: \bigcirc
leading coefficient: \bigcirc

Explanation:

Step1: Verify polynomial criteria

A polynomial function has non-negative integer exponents on the variable, and all coefficients are real numbers. For $p(x) = \frac{1}{2}x^2 + 3x - 4x^3 + 6x^4 - 1$, all exponents of $x$ (2, 1, 3, 4, 0) are non-negative integers, and all coefficients are real numbers. So it is a polynomial function.

Step2: Rearrange into standard form

Standard form of a polynomial orders terms from highest to lowest degree of $x$.
$$p(x) = 6x^4 - 4x^3 + \frac{1}{2}x^2 + 3x - 1$$

Step3: Identify degree

The degree is the highest exponent of $x$, which is 4.

Step4: Identify polynomial type

A 4th-degree polynomial is called a quartic polynomial.

Step5: Identify leading coefficient

The leading coefficient is the coefficient of the highest-degree term, which is 6.

Answer:

polynomial function
$p(x) = 6x^4 - 4x^3 + \frac{1}{2}x^2 + 3x - 1$
degree: 4
type: quartic polynomial
leading coefficient: 6