Given the x-intercepts J(-1; 0), K((1)/(2); 0), and L(3; 0), the factors of f(x) are (x+1), (x-(1)/(2)), and (x-3).

Hey asanda, good to see you again. Step 1: Use the x-intercepts to form the factored equation. Given the x-intercepts J(-1; 0), K((1)/(2); 0), and L(3; 0), the factors of f(x) are (x+1), (x-(1)/(2)), and (x-3). The general form of a cubic function is f(x) = a(x-x_1)(x-x_2)(x-x_3). From the given equation f(x) = 2x^3 + bx^2 + cx + d, the leading coefficient a=2. So, we can write f(x) as: f(x) = 2(x+1)(x-(1)/(2))(x-3) Step 2: Expand the first two factors. Multiply (x+1) by (x-(1)/(2)): (x+1)(x-(1)/(2)) = x^2 - (1)/(2)x + x - (1)/(2) = x^2 + (1)/(2)x - (1)/(2) Step 3: Multiply the result by the third factor (x-3). (x^2 + (1)/(2)x - (1)/(2))(x-3) = x^3 - 3x^2 + (1)/(2)x^2 - (3)/(2)x - (1)/(2)x + (3)/(2) Combine like terms: = x^3 + (-3 + (1)/(2))x^2 + (-(3)/(2) - (1)/(2))x + (3)/(2) = x^3 - (5)/(2)x^2 - 2x + (3)/(2) Step 4: Multiply by the leading coefficient 2 and compare with the given equation. f(x) = 2(x^3 - (5)/(2)x^2 - 2x + (3)/(2)) f(x) = 2x^3 - 5x^2 - 4x + 3 Comparing this with f(x) = 2x^3 + bx^2 + cx + d, we find the values of b, c, and d: b = -5 c = -4 d = 3 The values are b = -5, c = -4, d = 3. Send me the next one 📸