https://file.aiquickdraw.com/mathbot/file/1727729854439k5itlbz2.jpg

The equation given in the problem is:

\[
x^3 - 4x^2 - 3x + 18 = 0
\]

The task is to identify the correct factored form of this cubic equation from the given options.

### Step 1: Identifying Possible Roots

We are told that "three" is a zero of the equation. So, \( x = 3 \) is one of the roots. Using this, we can perform synthetic or polynomial division to factor out \( (x - 3) \) from the cubic polynomial.

### Step 2: Polynomial Division
Using synthetic division to divide \( x^3 - 4x^2 - 3x + 18 \) by \( (x - 3) \):

\[
\begin{array}{r|rrrr}
3 & 1 & -4 & -3 & 18 \\
  &   & 3  & -3 & -18 \\
\hline
  & 1 & -1 & -6 & 0 \\
\end{array}
\]

The result of the division is \( x^2 - x - 6 \).

### Step 3: Factoring \( x^2 - x - 6 \)

Now, we factor \( x^2 - x - 6 \):

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

### Step 4: Factored Form of the Cubic Equation

Thus, the factored form of the cubic equation is:

\[
(x - 3)(x - 3)(x + 2) = (x - 3)^2(x + 2)
\]

### Conclusion

The correct factored form is:

\[
(x - 3)^2(x + 2) = 0
\]

This matches the third option provided in the problem. Therefore, the correct answer is:

\[
\boxed{(x - 3)^2(x + 2) = 0}
\]

---

Do you want more details on any of the steps or have any other questions?

Here are five related questions:
1. What are the steps involved in performing synthetic division?
2. How can we determine the roots of a cubic equation using factoring methods?
3. What is the relationship between the roots of an equation and its factors?
4. Can every cubic polynomial be factored into linear terms? Why or why not?
5. How does the quadratic formula help in factoring quadratic expressions?

**Tip:** When factoring cubic equations, always look for potential integer roots using synthetic division or trial and error!

There is a zero of the equation x^3 - 4x^2 - 3x + 18 = 0. Which factored form is equivalent to the equation?

Learn how to factor the cubic equation x^3 - 4x^2 - 3x + 18. The solution involves synthetic division and factoring methods to find that (x - 3)^2(x + 2) = 0. This algebra problem is suitable for students in Grades 10-12.

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