Find the \( x \)-and \( y \)-intercepts of the rational function. (If an answer does not exist, enter DNE.) \[ t(x)=\frac{x^{2}-3 x-10}{x-8} \] \( x \)-intercept (smaller \( x \)-value) \( \quad(x, y)=(\square) \) \( x \)-intercept (larger \( x \)-value) \( \quad(x, y)=(x, y)=(\square) \) \( y \)-intercept

To find the \( x \)- and \( y \)-intercepts of the rational function \[ t(x) = \frac{x^2 - 3x - 10}{x - 8}, \] we will follow these steps: ### Finding the \( x \)-intercepts The \( x \)-intercepts occur where \( t(x) = 0 \). This happens when the numerator is equal to zero (as long as the denominator is not zero at those points). 1. Set the numerator equal to zero: \[ x^2 - 3x - 10 = 0. \] 2. Factor the quadratic: \[ (x - 5)(x + 2) = 0. \] 3. Solve for \( x \): \[ x - 5 = 0 \quad \Rightarrow \quad x = 5, \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2. \] Thus, the \( x \)-intercepts are \( (-2, 0) \) and \( (5, 0) \). ### Finding the \( y \)-intercept The \( y \)-intercept occurs where \( x = 0 \). 1. Substitute \( x = 0 \) into the function: \[ t(0) = \frac{0^2 - 3(0) - 10}{0 - 8} = \frac{-10}{-8} = \frac{10}{8} = \frac{5}{4}. \] Thus, the \( y \)-intercept is \( (0, \frac{5}{4}) \). ### Summary of Intercepts - \( x \)-intercept (smaller \( x \)-value): \( (x, y) = (-2, 0) \) - \( x \)-intercept (larger \( x \)-value): \( (x, y) = (5, 0) \) - \( y \)-intercept: \( (0, \frac{5}{4}) \) So, the final answers are: - \( x \)-intercept (smaller \( x \)-value): \( (-2, 0) \) - \( x \)-intercept (larger \( x \)-value): \( (5, 0) \) - \( y \)-intercept: \( (0, \frac{5}{4}) \)