Determine Polynomial Degree and Standard Form of g(x) = (4

Math Problem Statement

Determine whether the following function is a polynomial function. If the function is a polynomial function, state its degree. If it is not, tell why not. Write the polynomial in standard form. Then identify the leading term and the constant term. g(x)equalsStartFraction 4 minus x cubed Over 3 EndFraction Question content area bottom Part 1 Determine whether g(x) is a polynomial or not. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. It is a polynomial of degree

enter your response here. (Type an integer or a fraction.) B. It is not a polynomial because the variable x is raised to the

enter your response here power, which is not a nonnegative integer. (Type an integer or a fraction.) C. It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomial in the denominator is of positive degree.

Solution

Let's analyze the given function:

$g(x) = \frac{4 - x^3}{3}$

This function can be rewritten as:

$g(x) = \frac{1}{3}(4 - x^3) = \frac{1}{3} \cdot 4 - \frac{1}{3} \cdot x^3 = \frac{4}{3} - \frac{1}{3}x^3$

Step 1: Check if it's a polynomial function

A polynomial function is an expression made up of terms, each consisting of a constant coefficient and a variable raised to a non-negative integer exponent. There must be no division by variables.

In this case, after simplifying the expression for $g(x)$ , the terms are:

$\frac{4}{3}$ , a constant term.
$-\frac{1}{3}x^3$ , a term with $x$ raised to a non-negative integer power (3).

There is no division by $x$ , and both terms involve constants or powers of $x$ that are non-negative integers. Therefore, this is a polynomial function.

Step 2: Identify the degree

The degree of a polynomial function is the highest power of the variable $x$ in the expression. In this case, the highest power of $x$ is $3$ .

Thus, the degree of this polynomial is 3.

Step 3: Write the polynomial in standard form

The standard form of a polynomial is written with the terms in decreasing order of the exponents of $x$ :

$g(x) = -\frac{1}{3}x^3 + \frac{4}{3}$

Step 4: Identify the leading term and the constant term

The leading term is the term with the highest power of $x$ , which is $-\frac{1}{3}x^3$ .
The constant term is the term without $x$ , which is $\frac{4}{3}$ .

Conclusion:

The function $g(x)$ is a polynomial of degree 3. The leading term is $-\frac{1}{3}x^3$ , and the constant term is $\frac{4}{3}$ .

Would you like further details on this or any related topic?

Here are 5 related questions:

What makes a function not a polynomial?
How do you determine the degree of more complex polynomial functions?
Can polynomial functions have negative exponents or fractional exponents?
How does the degree of a polynomial affect its graph's behavior?
What are some examples of non-polynomial functions?

Tip: Always ensure that variables in a polynomial are raised to non-negative integer exponents and not in the denominator.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Degree of a Polynomial
Standard Form of a Polynomial

Formulas

Polynomial in standard form: a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0

Theorems

Definition of a Polynomial Function

Suitable Grade Level

Grades 9-11

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