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.