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The given function is:

\[
f(x) = x^3 + 2x^2 - 15x - 36
\]

It appears that \( x = -3 \) is one of the roots (since it is already filled in). To find the remaining roots, we can use synthetic division or factor the cubic equation.

### Step 1: Synthetic Division
We'll perform synthetic division with \( x = -3 \) to reduce the cubic polynomial to a quadratic one.

The coefficients of \( f(x) \) are: \( 1, 2, -15, -36 \).

Using synthetic division:

\[
\begin{array}{r|rrrr}
-3 & 1  & 2   & -15 & -36  \\
   &    & -3  & 3   & 36   \\
\hline
   & 1  & -1  & -12 & 0    \\
\end{array}
\]

The quotient is \( x^2 - x - 12 \).

### Step 2: Factor the Quadratic
Now, factor the quadratic:

\[
x^2 - x - 12 = (x - 4)(x + 3)
\]

So, the remaining roots are \( x = 4 \) and \( x = -3 \).

### Step 3: List All the Roots
Since \( x = -3 \) is a double root, we list it twice.

Thus, the zeros of the function are:

\[
x = -3, -3, 4
\]

Would you like further details on this process or have any questions?

Here are five related questions:
1. How is synthetic division used to find roots of polynomials?
2. What is the difference between simple and double roots?
3. How can you verify if a root is correct for a given polynomial?
4. What are the other methods for solving cubic equations besides factoring?
5. How does the Factor Theorem relate to solving polynomial equations?

**Tip:** When you find a root of a cubic equation, always divide the cubic by \( x - \text{root} \) to reduce the degree and make further solving easier.

Find all the zeros of f(x) = x^3 + 2x^2 - 15x - 36.

Learn how to find all zeros of the cubic function f(x) = x^3 + 2x^2 - 15x - 36 using synthetic division and factoring. This problem covers algebraic concepts such as polynomial equations, factoring, and synthetic division. Suitable for students in Grades 10-12.

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