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The problem states:

"The function \( f : f(x) = x^2 (x - 3)^2 \) is a polynomial function of the ... degree." with the options:
- (a) first
- (b) second
- (c) third
- (d) fourth

To determine the degree of the polynomial, let's expand and simplify the expression.

Given:
\[
f(x) = x^2 (x - 3)^2
\]

### Step 1: Expand \((x - 3)^2\)
\[
(x - 3)^2 = x^2 - 6x + 9
\]

### Step 2: Multiply by \(x^2\)
\[
f(x) = x^2 (x^2 - 6x + 9) = x^4 - 6x^3 + 9x^2
\]

### Step 3: Determine the Degree
The highest power of \(x\) in the polynomial is \(x^4\). Therefore, the degree of the polynomial is **fourth**.

So, the correct answer is:
**(d) fourth**

Would you like a detailed explanation of the solution or have any other questions?

Here are 5 related questions you may find interesting:
1. How do you determine the degree of a polynomial from its standard form?
2. What is the difference between the degree and the leading coefficient of a polynomial?
3. Can a polynomial have a degree of zero? If yes, what does it represent?
4. How would the degree change if the function was \(x^3(x - 3)^2\)?
5. What are the possible shapes of the graph for a fourth-degree polynomial?

**Tip:** Always check the highest power of \(x\) after expanding all terms to find the degree of a polynomial.

The function f(x) = x^2 (x - 3)^2 is a polynomial function of the ... degree. Options: (a) first, (b) second, (c) third, (d) fourth.

Learn how to determine the degree of the polynomial f(x) = x^2(x - 3)^2 by expanding the terms and finding the highest power of x. This example helps students in Grades 9-12 understand polynomial degrees and algebraic expansions.

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