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)equals3 left parenthesis x minus 1 right parenthesis squared left parenthesis x squared plus 2 right parenthesis
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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.

To determine whether the function \( G(x) = 3(x - 1)^2 (x^2 + 2) \) is a polynomial and, if so, to find its degree, let's break the function down:

### Step 1: Expand the function

We need to expand \( G(x) = 3(x - 1)^2(x^2 + 2) \).

- First, expand \( (x - 1)^2 \):

\[
(x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1
\]

Now the function becomes:

\[
G(x) = 3(x^2 - 2x + 1)(x^2 + 2)
\]

- Next, expand \( (x^2 - 2x + 1)(x^2 + 2) \) by distributing each term:

\[
(x^2 - 2x + 1)(x^2 + 2) = x^2(x^2 + 2) - 2x(x^2 + 2) + 1(x^2 + 2)
\]
\[
= x^4 + 2x^2 - 2x^3 - 4x + x^2 + 2
\]
\[
= x^4 - 2x^3 + 3x^2 - 4x + 2
\]

Now multiply by the 3:

\[
G(x) = 3(x^4 - 2x^3 + 3x^2 - 4x + 2) = 3x^4 - 6x^3 + 9x^2 - 12x + 6
\]

### Step 2: Check if \( G(x) \) is a polynomial

A function is a polynomial if it consists of terms with non-negative integer powers of \( x \) and has no variables in the denominator or exponents that are not whole numbers.

The expanded form of \( G(x) \) is \( G(x) = 3x^4 - 6x^3 + 9x^2 - 12x + 6 \), which consists only of terms with non-negative integer powers of \( x \). Therefore, \( G(x) \) is a polynomial.

### Step 3: Determine the degree, leading term, and constant term

- **Degree**: The degree of a polynomial is the highest power of \( x \). In this case, the highest power is \( x^4 \), so the degree is 4.
- **Leading term**: The leading term is the term with the highest degree. Here, it is \( 3x^4 \).
- **Constant term**: The constant term is the term without any \( x \). Here, it is 6.

### Conclusion:

- \( G(x) \) is a polynomial of degree 4.
- The polynomial in standard form is \( G(x) = 3x^4 - 6x^3 + 9x^2 - 12x + 6 \).
- The leading term is \( 3x^4 \).
- The constant term is 6.

### Final Answer:
A. It is a polynomial of degree 4.

Would you like further details on this? Here are some related questions:

1. What makes a function non-polynomial in general?
2. Can a polynomial have fractional or negative exponents?
3. What are the characteristics of polynomials of even vs. odd degree?
4. How does the leading coefficient affect the graph of a polynomial function?
5. Can the degree of a polynomial be determined by looking at factored form?

**Tip:** Expanding a polynomial helps you see all its terms clearly, making it easier to identify degree, leading, and constant terms.

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) = 3(x - 1)^2(x^2 + 2)


Learn how to determine if the function G(x) = 3(x - 1)^2(x^2 + 2) is a polynomial and find its degree. Follow step-by-step expansion to write the polynomial in standard form and identify the leading and constant terms. Suitable for students in Grades 10-12.

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