Find the \( x \)-intercepts and \( y \)-intercept of the following function. \[ f(x)=(x-4)(x+1)(x+8) \]

To find the \( x \)-intercepts and the \( y \)-intercept of the function \( f(x) = (x-4)(x+1)(x+8) \), follow these steps: ### \( x \)-Intercepts The \( x \)-intercepts occur where \( f(x) = 0 \). Set the function equal to zero and solve for \( x \): \[ (x - 4)(x + 1)(x + 8) = 0 \] This equation is satisfied when any of the factors is zero: 1. \( x - 4 = 0 \) \(\Rightarrow\) \( x = 4 \) 2. \( x + 1 = 0 \) \(\Rightarrow\) \( x = -1 \) 3. \( x + 8 = 0 \) \(\Rightarrow\) \( x = -8 \) So, the \( x \)-intercepts are: - \( (4, 0) \) - \( (-1, 0) \) - \( (-8, 0) \) ### \( y \)-Intercept The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function: \[ f(0) = (0 - 4)(0 + 1)(0 + 8) = (-4)(1)(8) = -32 \] So, the \( y \)-intercept is: - \( (0, -32) \) ### Summary - **\( x \)-intercepts:** \( (4, 0) \), \( (-1, 0) \), and \( (-8, 0) \) - **\( y \)-intercept:** \( (0, -32) \)