This quiz: 100 point(s) 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. \[ f(x)=2(x-7)(x+4)^{3} \] Determine the zero(s). The zero(s) is/are \( \square \) \( \square \). (Type integers or decimals. Use a comma to separate answers as needed.) Determine the multiplicities of the zero(s). Select the correct choice below and, if necessary, fill in the answer box(es) within your choice. A. There are two zeros. The multiplicity of the largest zero is \( \square \) The multiplicity of the smallest zero is \( \square \) ]. (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 \( \square \) 1. (Simplify your answer.)

Let's analyze the polynomial function: \[ f(x) = 2(x - 7)(x + 4)^3 \] ### 1. **Finding the Zero(s)** To find the zeros of the function, set \( f(x) = 0 \): \[ 2(x - 7)(x + 4)^3 = 0 \] This equation is satisfied when either \( x - 7 = 0 \) or \( (x + 4)^3 = 0 \). - **Zero 1:** \( x - 7 = 0 \) → \( x = 7 \) - **Zero 2:** \( x + 4 = 0 \) → \( x = -4 \) **Zero(s):** \( 7, -4 \) ### 2. **Determining Multiplicities** Each zero's multiplicity is determined by the exponent of its corresponding factor in the polynomial. - **Zero at \( x = 7 \):** The factor \( (x - 7) \) has an exponent of **1**. - **Zero at \( x = -4 \):** The factor \( (x + 4)^3 \) has an exponent of **3**. **Multiplicity:** - **Largest Zero (\( 7 \)):** Multiplicity **1** - **Smallest Zero (\( -4 \)):** Multiplicity **3** **Selection:** **A. There are two zeros. The multiplicity of the largest zero is \( 1 \). The multiplicity of the smallest zero is \( 3 \).** ### 3. **Behavior of the Graph at Each Zero** - **Zero at \( x = 7 \) (Multiplicity 1):** - **Behavior:** The graph **crosses** the \( x \)-axis. - **Zero at \( x = -4 \) (Multiplicity 3):** - **Behavior:** The graph **crosses** the \( x \)-axis. *Note:* Since both zeros have **odd** multiplicities, the graph will **cross** the \( x \)-axis at each zero. ### **Summary** - **Zero(s):** \( 7, -4 \) - **Multiplicity of \( 7 \):** \( 1 \) (Graph crosses the \( x \)-axis) - **Multiplicity of \( -4 \):** \( 3 \) (Graph crosses the \( x \)-axis)