Write a polynomial function in standard form with leading coefficients of 2 and zeros at x= -2, 0 and 3i

To write a polynomial function in standard form with leading coefficients of 2 and zeros at \( x = -2 \), \( x = 0 \), and \( x = 3i \), follow these steps:

### Step 1: Express the zeros as factors.
- For the zero \( x = -2 \), the factor is \( (x + 2) \).
- For the zero \( x = 0 \), the factor is \( x \).
- For the zero \( x = 3i \), the factor is \( (x - 3i) \).

### Step 2: Consider the complex conjugate.
- Since the coefficients of the polynomial must be real numbers, the complex zero \( x = 3i \) implies the conjugate \( x = -3i \) must also be a zero.
- The corresponding factor is \( (x + 3i) \).

### Step 3: Multiply the factors.
The polynomial function can be written as:
\[
f(x) = 2x(x + 2)(x - 3i)(x + 3i)
\]

### Step 4: Simplify the expression.
First, simplify the product of the complex conjugates:
\[
(x - 3i)(x + 3i) = x^2 - (3i)^2 = x^2 - (-9) = x^2 + 9
\]
Now, multiply all the factors together:
\[
f(x) = 2x(x + 2)(x^2 + 9)
\]

### Step 5: Expand the polynomial.
Multiply \( (x + 2) \) with \( (x^2 + 9) \):
\[
(x + 2)(x^2 + 9) = x(x^2 + 9) + 2(x^2 + 9) = x^3 + 9x + 2x^2 + 18
\]
Now multiply by \( 2x \):
\[
f(x) = 2x(x^3 + 2x^2 + 9x + 18)
\]
\[
f(x) = 2x^4 + 4x^3 + 18x^2 + 36x
\]

### Final Answer:
The polynomial function in standard form is:
\[
f(x) = 2x^4 + 4x^3 + 18x^2 + 36x
\]

Would you like more details on the steps? Here are 5 related questions you could explore:

1. How do you determine the leading coefficient of a polynomial?
2. What is the significance of including the complex conjugate in the polynomial?
3. How do you factor a polynomial given its zeros?
4. What are some common mistakes to avoid when expanding polynomials?
5. How can you verify the roots of a polynomial after constructing it?

**Tip:** When dealing with complex roots, always ensure to include the complex conjugate to maintain real coefficients in your polynomial.

Learn how to construct a polynomial function with a leading coefficient of 2 and zeros at x = -2, 0, and 3i. This problem involves polynomial expansion, the use of complex conjugate pairs, and the Fundamental Theorem of Algebra. Suitable for students in Grades 10-12.

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