Let's analyze the function $G(x) = 3(x - 4)^2 (x^2 + 1)$ and determine if it is a polynomial.

Step 1: Check if $G(x)$ is a polynomial

A polynomial is a function of the form:
$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ where $n$ is a nonnegative integer, and all coefficients $a_i$ are real numbers.

• The function involves terms of $(x - 4)^2$ and $(x^2 + 1)$, both of which are polynomials.
• Multiplying two polynomials always results in another polynomial.
• There are no denominators, fractional powers, or negative exponents here.

Thus, $G(x)$ is a polynomial function.

Step 2: Determine the degree of the polynomial

To find the degree, expand $G(x)$ step by step:

1. $(x - 4)^2$ expands to: $(x - 4)^2 = x^2 - 8x + 16$

2. Multiply this result by $(x^2 + 1)$: $G(x) = 3(x^2 - 8x + 16)(x^2 + 1)$

   Distribute $(x^2 + 1)$ across $(x^2 - 8x + 16)$: $(x^2 - 8x + 16)(x^2 + 1) = x^2 \cdot x^2 + x^2 \cdot 1 - 8x \cdot x^2 - 8x \cdot 1 + 16 \cdot x^2 + 16 \cdot 1$

   Simplify term by term: $x^4 + x^2 - 8x^3 - 8x + 16x^2 + 16$

   Combine like terms: $x^4 - 8x^3 + 17x^2 - 8x + 16$

3. Multiply by 3: $G(x) = 3(x^4 - 8x^3 + 17x^2 - 8x + 16)$

   Simplify: $G(x) = 3x^4 - 24x^3 + 51x^2 - 24x + 48$

Step 3: Identify the degree, leading term, and constant term

• Degree: The highest power of $x$ is $4$.
• Leading term: $3x^4$ (coefficient is $3$).
• Constant term: $48$.

Final Answers:

1. $G(x)$ is a polynomial.
2. Degree: $4$.
3. Leading term: $3x^4$.
4. Constant term: $48$.

Selection:

• First Part: $A.$ It is a polynomial of degree 4.
• Second Part: $A.$ The polynomial in standard form is:
  $G(x) = 3x^4 - 24x^3 + 51x^2 - 24x + 48$
  with leading term $3x^4$ and constant $48$.

Do you need more clarification or expansion on any step? Let me know!

Related Questions:

1. What is the degree of a product of two polynomials in general?
2. How do you determine if a function is not a polynomial?
3. How would you simplify $(x - a)^2(x + b)$?
4. What happens to the degree of a polynomial when multiplied by a constant?
5. How can you expand $(x^2 + a)(x^2 + b)$?

Tip:

Always ensure all terms in a polynomial have nonnegative integer powers of $x$ and check for no divisions by $x$ or roots!