Which of the following is an x-intercept of the function, \( f(x)=x^{3}+3 x^{2}-10 x-24 \) ? A. 4 B. 2 c. -4 D. -3

To find the x-intercept of the function \( f(x) = x^{3} + 3x^{2} - 10x - 24 \), we need to set \( f(x) = 0 \). This means we want to determine which of the provided options makes the equation true. By substituting each option into the function: - For A (4), \( f(4) = 4^3 + 3(4^2) - 10(4) - 24 = 64 + 48 - 40 - 24 = 48 \neq 0 \). - For B (2), \( f(2) = 2^3 + 3(2^2) - 10(2) - 24 = 8 + 12 - 20 - 24 = -24 \neq 0 \). - For C (-4), \( f(-4) = (-4)^3 + 3(-4)^2 - 10(-4) - 24 = -64 + 48 + 40 - 24 = 0 \). - For D (-3), \( f(-3) = (-3)^3 + 3(-3)^2 - 10(-3) - 24 = -27 + 27 + 30 - 24 = 6 \neq 0 \). So the correct answer is C: -4. One can easily verify this result graphically using a graphing calculator or software, which should show the curve crossing the x-axis at -4. Remember to always check your calculations for any potential errors, as it's easy to mix up signs when substituting values. Accurate arithmetic is key to finding the correct intercepts!