Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the graph crosses the x -axis or touches the x -axis and turns around at each zero. $$ \mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+4 \mathrm{x}^{2}-9 \mathrm{x}-36 $$ Determine the zero(s), if they exist. The zero(s) is/are $$ -4,-3,3 $$. (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s), if they exist. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There are three zeros. The multiplicity of the smallest zero is The multiplicity of the other zero ish (Simplify your answers.) B. There is one zero. The multiplicicity of the zero is (Simplify your answer.) C. There are two zeros. The multiplicity of the smallest zero is $$ \square $$. The multiplicity of the largest zero is (Simplify your answers.)

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UpStudy AI · Accepted Answer

To analyze the polynomial function $$ f(x) = x^3 + 4x^2 - 9x - 36 $$, we'll determine its zeros, their multiplicities, and the behavior of the graph at each zero.

### **Step 1: Finding the Zeros**

Using the Rational Root Theorem, possible rational zeros are factors of the constant term divided by factors of the leading coefficient. For $$ f(x) $$, the possible rational roots are:
$$ \pm1, \pm2, \pm3, \pm4, \pm6, \pm9, \pm12, \pm18, \pm36 $$

Testing these, we find:

- **$$ x = -4 $$:**
  $$
  f(-4) = (-4)^3 + 4(-4)^2 - 9(-4) - 36 = -64 + 64 + 36 - 36 = 0
  $$
  So, $$ x = -4 $$ is a zero.

Using synthetic division to factor out $$ (x + 4) $$:

$$
\begin{array}{r|rrrr}
-4 & 1 & 4 & -9 & -36 \
   &   & -4 & 0 & 36 \
\hline
   & 1 & 0 & -9 & 0 \
\end{array}
$$

The quotient is $$ x^2 - 9 $$, which factors further:
$$
x^2 - 9 = (x - 3)(x + 3)
$$

Thus, the zeros are:
$$
x = -4, \quad x = -3, \quad x = 3
$$

### **Step 2: Determining Multiplicities**

Each zero comes from a distinct linear factor:
$$
f(x) = (x + 4)(x - 3)(x + 3)
$$
Therefore, each zero has a **multiplicity of 1**.

### **Step 3: Graph Behavior at Each Zero**

- **Multiplicity 1 (Odd Multiplicity):** The graph **crosses** the x-axis at each zero.

### **Summary of Findings**

- **Zeros:** $$ -4, -3, 3 $$
- **Multiplicities:** Each zero has a multiplicity of **1**.
- **Graph Behavior:** The graph **crosses** the x-axis at each zero.

### **Answer Selection**

Based on the options provided:

**A. There are three zeros. The multiplicity of the smallest zero is 1. The multiplicity of the other zeros is 1.**

This accurately reflects the multiplicities and the number of zeros.

**Final Answer:**

**A.** There are three zeros. The multiplicity of the smallest zero is 1. The multiplicity of the other zeros is 1. The graph crosses the x-axis at each zero.
The zeros are -4, -3, and 3, each with a multiplicity of 1. The graph crosses the x-axis at each zero.

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