We are given the polynomial $f(x) = 9x^4 - 12x^3 - 38x^2 - 20x - 3$ and we know that it has a zero of multiplicity 2 at $x = -\frac{1}{3}$. To find the remaining real zeros, we can use synthetic division with $x + \frac{1}{3}$ as a divisor.

Step 1: Synthetic Division

We will perform synthetic division twice, because the zero has a multiplicity of 2, which means we must divide by $x + \frac{1}{3}$ two times.

1. First Division:

   • The coefficients of the polynomial $9x^4 - 12x^3 - 38x^2 - 20x - 3$ are: $[9, -12, -38, -20, -3]$.
   • We divide by $x + \frac{1}{3}$ (which corresponds to using $-\frac{1}{3}$ in synthetic division).

2. Second Division:

   • Once we get the result from the first division, we repeat the process to divide again by $x + \frac{1}{3}$.

Let's perform the division now.After performing synthetic division twice, the quotient we obtain is $9x^2 - 18x - 27$, with a remainder of 0, indicating the division was successful.

Step 2: Factor the Quotient

Now, we need to factor $9x^2 - 18x - 27$. First, factor out the common factor of 9:

$9(x^2 - 2x - 3)$

Next, we factor $x^2 - 2x - 3$ by finding two numbers that multiply to -3 and add to -2. These numbers are -3 and 1:

$9(x - 3)(x + 1)$

Step 3: Factored Form of the Polynomial

The factored form of the original polynomial $f(x)$ is:

$f(x) = 9(x + \frac{1}{3})^2(x - 3)(x + 1)$

Step 4: Remaining Real Zeros

The remaining real zeros of the polynomial are $x = 3$ and $x = -1$.

Final Answer:

• Remaining Zeros: $3, -1$
• Factored Form: $f(x) = 9(x + \frac{1}{3})^2(x - 3)(x + 1)$

Do you need further details on the steps, or have any questions?

Here are some related questions to deepen your understanding:

1. What is synthetic division, and how is it different from long division?
2. How do you determine the multiplicity of a zero in a polynomial?
3. Why is it important that the remainder is 0 when using synthetic division with a known zero?
4. Can you factor a quadratic equation without factoring out a common term first?
5. How does the factored form of a polynomial help in graphing its function?

Tip: When factoring polynomials, always check for common factors first to simplify your work.