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

Consider the following polynomial function.

f(x) = (x + 3)^2(x - 2)^3(x - 1)

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

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

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none 1 2 3 4

Solution

To find the \( x \)-intercepts at which the function \( f(x) \) crosses the axis, we’ll analyze the given polynomial function step by step. Given: \[ f(x) = (x + 3)^2(x - 2)^3(x - 1) \] Step 1: Identify the roots of the polynomial Each factor of the polynomial gives a root where the function intersects the \( x \)-axis. \[ x + 3 = 0 \implies x = -3 & \] \[ x - 2 = 0 \implies x = 2 & \] \[ x - 1 = 0 \implies x = 1 & \] Step 2: Determine the multiplicity of each root - \( x = -3 \) has a multiplicity of 2. - \( x = 2 \) has a multiplicity of 3. - \( x = 1 \) has a multiplicity of 1. Step 3: Analyze the behavior at each root - Even multiplicity (e.g., 2): The graph touches the \( x \)-axis but does not cross it. At \( x = -3 \): The graph does not cross the axis. - Odd multiplicity (e.g., 1, 3): The graph crosses the \( x \)-axis. At \( x = 2 \) and \( x = 1 \): The graph crosses the axis. Step 4: Summarize the \( x \)-intercepts where the graph crosses the axis \[ (2, 0) \quad \text{and} \quad (1, 0) \] Conclusion: There are 2 \( x \)-intercepts where the function \( f(x) \) crosses the \( x \)-axis.