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To form a polynomial \( f(x) \) with real coefficients given the specified degree and zeros, let's follow these steps based on the information provided:

### Given Information:
- Degree: 4
- Zeros: \( 3 + 3i \) and \( 3 \) with multiplicity 2

### Step 1: Understand Complex Conjugate Pairs
Since the polynomial has real coefficients, if \( 3 + 3i \) is a zero, then its complex conjugate \( 3 - 3i \) must also be a zero.

### Step 2: List all Zeros
The zeros of the polynomial are:
- \( 3 + 3i \)
- \( 3 - 3i \)
- \( 3 \) (with multiplicity 2)

### Step 3: Form the Polynomial
The polynomial can be expressed as the product of factors corresponding to its zeros:
\[
f(x) = a(x - (3 + 3i))(x - (3 - 3i))(x - 3)^2
\]

### Step 4: Multiply the Factors
1. Multiply the complex conjugate factors:
   \[
   (x - (3 + 3i))(x - (3 - 3i)) = (x - 3 - 3i)(x - 3 + 3i)
   \]
   This simplifies using the difference of squares formula:
   \[
   (x - 3)^2 - (3i)^2 = (x - 3)^2 + 9
   \]
   Expanding \((x - 3)^2\):
   \[
   (x - 3)^2 = x^2 - 6x + 9
   \]
   So, we have:
   \[
   (x - 3)^2 + 9 = x^2 - 6x + 9 + 9 = x^2 - 6x + 18
   \]

2. Now, combine this result with the other factor \((x - 3)^2\):
   \[
   f(x) = a(x^2 - 6x + 18)(x - 3)^2
   \]
3. Expand \((x - 3)^2\) again:
   \[
   (x - 3)^2 = x^2 - 6x + 9
   \]

4. Multiply \( (x^2 - 6x + 18)(x^2 - 6x + 9) \) to get the expanded form:
   \[
   f(x) = a(x^4 - 12x^3 + 63x^2 - 108x + 162)
   \]

### Final Answer
The polynomial \( f(x) \) in its expanded form is:
\[
f(x) = a(x^4 - 12x^3 + 63x^2 - 108x + 162)
\]

Let me know if you would like more details or have any questions!

### Relative Questions:
1. How do you determine the conjugate of a complex number?
2. What is the significance of the multiplicity of a root in a polynomial?
3. How do you expand expressions using the distributive property?
4. What is the difference between real and complex roots in a polynomial?
5. How can the degree of a polynomial affect its possible number of roots?

### Tip:
Always remember that if a polynomial has real coefficients, complex roots must come in conjugate pairs.

Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree 4; zeros: 3 + 3i; 3 multiplicity 2.

This problem explores how to form a polynomial with real coefficients, given a degree of 4, zeros including 3 + 3i and 3 with multiplicity 2. The solution uses polynomial factorization and the complex conjugate root theorem. Suitable for students in Grades 10-12.

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