Questions: f(x)=4 x(x-3)^2(x+6)^3

Solution Steps

To analyze the function \( f(x) = 4x(x-3)^2(x+6)^3 \), we can follow these steps:

1. Identify the roots: Set \( f(x) = 0 \) and solve for \( x \).
2. Determine the multiplicity of each root: The exponents in the factors indicate the multiplicity.
3. Analyze the behavior around each root: Use the multiplicity to determine if the graph touches or crosses the x-axis at each root.

To find the roots of the function \( f(x) = 4x(x-3)^2(x+6)^3 \), we set \( f(x) = 0 \). The roots are determined to be: \[ x = -6, \quad x = 0, \quad x = 3 \]

The multiplicity of each root is derived from the factors of the function:

• For \( x = -6 \), the factor \( (x + 6)^3 \) indicates a multiplicity of \( 3 \).
• For \( x = 0 \), the factor \( x \) indicates a multiplicity of \( 1 \).
• For \( x = 3 \), the factor \( (x - 3)^2 \) indicates a multiplicity of \( 2 \).

Thus, the multiplicities are: \[ \text{Multiplicity of } x = -6: 3, \quad \text{Multiplicity of } x = 0: 1, \quad \text{Multiplicity of } x = 3: 2 \]

• At \( x = -6 \) (multiplicity \( 3 \)): The graph crosses the x-axis.
• At \( x = 0 \) (multiplicity \( 1 \)): The graph crosses the x-axis.
• At \( x = 3 \) (multiplicity \( 2 \)): The graph touches the x-axis and does not cross.

Final Answer