Question: Consider the following polynomial function. \[ f(x) = (x + 5)^2(x - 1)^3(x - 2) \] Step 2 of 3:…

Consider the following polynomial function.

\[ f(x) = (x + 5)^2(x - 1)^3(x - 2) \]

Step 2 of 3: Find the x-intercept(s) at which \( f \) crosses the axis. Express the intercept(s) as ordered pair(s).

Answer

Select the number of x-intercept(s) at which \( f \) crosses the axis.

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Solution

Step 1: Find the x-intercepts by setting \( f(x) = 0 \). \[ f(x) = (x + 5)^2(x - 1)^3(x - 2) = 0 & \] Step 2: Solve for \( x \). \[ x + 5 = 0 \\ x - 1 = 0 \\ x - 2 = 0 & \] \[ x = -5 \\ x = 1 \\ x = 2 & \] Step 3: Determine the multiplicity of each root to see if the graph crosses the x-axis. \[ \text{Multiplicity of } x = -5 \text{ is } 2 \\ \text{Multiplicity of } x = 1 \text{ is } 3 \\ \text{Multiplicity of } x = 2 \text{ is } 1 & \] Step 4: Analyze the multiplicity to determine the behavior at each x-intercept. \[ \text{Even multiplicity: Touches the axis} \\ \text{Odd multiplicity: Crosses the axis} & \] \[ x = -5 \text{ (even)} \rightarrow \text{Does not cross} \\ x = 1 \text{ (odd)} \rightarrow \text{Crosses} \\ x = 2 \text{ (odd)} \rightarrow \text{Crosses} & \] Final Answer: The x-intercepts where \( f \) crosses the axis are: \[ (1, 0) \\ (2, 0) & \]