Question 1 of 10 possible This question: 10 point(s) possible Submit quiz 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. \[ f(x)=2(x-7)(x+4)^{3} \] answer box(es) within your choice. A. There are two zeros. The multiplicity of the largest zero is 7 . The multiplicity of the smallest zero is \( \square \) 1. (Simplify your answers.) B. There are three zeros. The multiplicity of the largest zero is \( \square \) The multiplicity of the smallest zero is \( \square \) . The multiplicity of the other zero is \( \square \) (Simplify your answers.) C. There is one zero. The multiplicity of the zero is \( \square \) . (Simplify your answer.) Determine the behavior of the function at each zero. Select the correct choice below and, if necessary, fill in the answer boxes within your choice. A. The graph crosses the \( x \)-axis at all zeros. B. The graph crosses the \( x \)-axis at \( x= \) \( \square \) and touches the x -axis and turns around at \( \mathrm{x}= \) \( \square \) . (Simplify your answers. Type integers or decimals. Use a comma to separate answers as needed.) C. The graph touches the \( x \)-axis and turns around at all zeros. Time Remaining: 01:32:39 Next

Let's analyze the polynomial function \( f(x) = 2(x - 7)(x + 4)^3 \) to find its zeros, their multiplicities, and the behavior of the graph at each zero. ### **1. Finding the Zeros and Their Multiplicities** To find the zeros of \( f(x) \), set the function equal to zero: \[ 2(x - 7)(x + 4)^3 = 0 \] This equation is satisfied when either factor is zero: 1. **\( x - 7 = 0 \)** - **Zero:** \( x = 7 \) - **Multiplicity:** 1 (since the factor \( (x - 7) \) is raised to the power of 1) 2. **\( (x + 4)^3 = 0 \)** - **Zero:** \( x = -4 \) - **Multiplicity:** 3 (since the factor \( (x + 4) \) is raised to the power of 3) **Summary of Zeros:** - **\( x = 7 \)** with **multiplicity 1** - **\( x = -4 \)** with **multiplicity 3** ### **2. Behavior of the Graph at Each Zero** The behavior of the graph at each zero depends on the multiplicity: - **Odd Multiplicity:** The graph **crosses** the x-axis at the zero. - **Even Multiplicity:** The graph **touches** the x-axis and **turns around** at the zero. Applying this to our zeros: 1. **\( x = 7 \)** with **multiplicity 1** (odd) - **Behavior:** The graph **crosses** the x-axis at \( x = 7 \). 2. **\( x = -4 \)** with **multiplicity 3** (odd) - **Behavior:** The graph **crosses** the x-axis at \( x = -4 \). ### **Final Answer** **Zeros and Their Multiplicities:** - **\( x = 7 \)** with multiplicity **1** - **\( x = -4 \)** with multiplicity **3** **Graph Behavior:** - The graph **crosses** the x-axis at both \( x = 7 \) and \( x = -4 \). --- **Matching the Provided Options:** Based on the analysis: - **For the Zeros and Multiplicities:** - **Option A** is the closest match if we correct the typo regarding the multiplicity of the largest zero. - **Corrected Option A:** - There are two zeros. - The multiplicity of the largest zero (\( x = 7 \)) is **1**. - The multiplicity of the smallest zero (\( x = -4 \)) is **3**. - **For the Graph Behavior:** - **Option A** correctly states that the graph crosses the x-axis at all zeros. **Final Selection:** **A.** - **There are two zeros.** - **Multiplicity of the largest zero (\( x = 7 \)) is** **1**. - **Multiplicity of the smallest zero (\( x = -4 \)) is** **3**. **Behavior:** - **The graph crosses the \( x \)-axis at all zeros.**