Final answer:
To solve the cubic equation, divide it by (x-4) to obtain a quadratic equation, then use the quadratic formula to find the remaining solutions.
Step-by-step explanation:
To find all solutions to the cubic equation 12x³-59x²+46x-8=0, given that x=4 is one solution, we can use polynomial division or synthetic division to factor the polynomial. Since we know that x=4 is a solution, (x-4) will be a factor of the polynomial. Therefore, we divide the polynomial by (x-4) to get a quadratic equation.
After performing the division, the quadratic equation we get is of the form ax² + bx + c = 0. To solve this quadratic equation for the remaining solutions, we can use the quadratic formula, which is x = (-b ± √(b² - 4ac))/(2a). Substituting the values of a, b, and c from the quadratic equation into the formula will give us the two other solutions of the cubic equation.