Question

determine whether $p(x) = \frac{1}{2}x^2 + 3x - 4x^3 + 6x^4 - 1$ is a polynomial function.

• polynomial function
• not a polynomial function

if so, write the polynomial function in standard form; otherwise leave the box empty.
$p(x) = \square$
state the degree, type, and leading coefficient of the polynomial function. if the function is not a polynomial function, answer
ot possible.\
degree $\square$
type $\square$
leading coefficient $\square$

Explanation:

Step1: Recall Polynomial Definition

A polynomial function is of the form \( p(x) = a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0 \), where \( n \) is a non - negative integer and \( a_i \) 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 for the constant term - 1) are non - negative integers, and coefficients (\(\frac{1}{2},3, - 4,6, - 1\)) are real numbers. So it is a polynomial function.

Step2: Write in Standard Form

Standard form of a polynomial is written in descending order of exponents of \( x \). Rearranging the terms of \( p(x)=\frac{1}{2}x^{2}+3x - 4x^{3}+6x^{4}-1 \), we get \( p(x)=6x^{4}-4x^{3}+\frac{1}{2}x^{2}+3x - 1 \).

Step3: Find Degree, Type, Leading Coefficient

• Degree: The degree of a polynomial is the highest power of \( x \) in the polynomial. In \( p(x)=6x^{4}-4x^{3}+\frac{1}{2}x^{2}+3x - 1 \), the highest power of \( x \) is 4, so the degree is 4.
• Type: A polynomial with degree 4 is called a quartic polynomial (or a 4th - degree polynomial).
• Leading Coefficient: The leading coefficient is the coefficient of the term with the highest degree. For the term \( 6x^{4} \), the coefficient is 6, so the leading coefficient is 6.

Answer:

• The function \( p(x) \) is a polynomial function.
• \( p(x)=\boldsymbol{6x^{4}-4x^{3}+\frac{1}{2}x^{2}+3x - 1} \)
• Degree: \(\boldsymbol{4}\)
• Type: \(\boldsymbol{\text{Quartic (4th - degree) polynomial}}\)
• Leading coefficient: \(\boldsymbol{6}\)