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The problem asks for the polynomial function of the lowest degree with real coefficients that has the zeros \( 0, -3, 1 + 3i \).

### Step-by-step solution:

1. **Given Zeros**: 
   The roots provided are \( 0 \), \( -3 \), and \( 1 + 3i \). Since the polynomial must have real coefficients and \( 1 + 3i \) is a complex number, its conjugate, \( 1 - 3i \), must also be a root.

2. **Forming the Polynomial**:
   The polynomial's factors will be:
   \[
   f(x) = x(x + 3)\left(x - (1 + 3i)\right)\left(x - (1 - 3i)\right)
   \]
   
3. **Simplifying the Complex Factors**:
   We can simplify the product of the two complex factors, using the difference of squares:
   \[
   \left(x - (1 + 3i)\right)\left(x - (1 - 3i)\right) = \left[(x - 1)^2 - (3i)^2\right] = (x - 1)^2 + 9 = x^2 - 2x + 10
   \]
   
4. **Expanding the Full Polynomial**:
   Now, we have the polynomial:
   \[
   f(x) = x(x + 3)(x^2 - 2x + 10)
   \]
   First, expand \( x(x + 3) \):
   \[
   x(x + 3) = x^2 + 3x
   \]
   Now expand the entire expression:
   \[
   f(x) = (x^2 + 3x)(x^2 - 2x + 10)
   \]
   Multiply term by term:
   \[
   f(x) = x^2(x^2 - 2x + 10) + 3x(x^2 - 2x + 10)
   \]
   \[
   f(x) = x^4 - 2x^3 + 10x^2 + 3x^3 - 6x^2 + 30x
   \]
   Combine like terms:
   \[
   f(x) = x^4 + (3x^3 - 2x^3) + (10x^2 - 6x^2) + 30x
   \]
   \[
   f(x) = x^4 + x^3 + 4x^2 + 30x
   \]

5. **Final Answer**:
   The polynomial is \( f(x) = x^4 + x^3 + 4x^2 + 30x \), which matches option **B**.

### The correct answer is B.

Would you like further clarification or additional details?

Here are 5 related questions:
1. How do you determine the degree of a polynomial from its roots?
2. Why do complex roots appear in conjugate pairs for polynomials with real coefficients?
3. What is the general form of a quadratic factor when given a complex root?
4. How do you expand polynomials using the distributive property?
5. Can a polynomial have more than one degree-4 form with the same set of zeros?

**Tip:** When working with complex roots, always remember that their conjugates must also be included to ensure the polynomial has real coefficients.

Find a polynomial function of the lowest degree with real coefficients that has the zeros of 0, -3, 1 + 3i.

Learn how to find the polynomial of the lowest degree with real coefficients that has the zeros 0, -3, and 1 + 3i. This step-by-step solution covers key concepts like complex roots, polynomial expansion, and the conjugate root theorem.

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