Final answer:
The equation has three solutions in the interval [-4, 4]: x = -4, x = 1, and x = -3.
Step-by-step explanation:
To show that the equation x³−13x+3=0 has three solutions in the interval [−4, 4], we can use the Intermediate Value Theorem. First, we find the values of f(x) = x³−13x+3 at the endpoints of the interval: f(-4) = -67 and f(4) = 41. Since f(-4) is negative and f(4) is positive, by the Intermediate Value Theorem, there must exist at least one solution between -4 and 4. To find the other two solutions, we can use the polynomial remainder theorem to factorize the equation. By synthetic division, we can find that (x+4) and (x-1) are factors of x³−13x+3=0. Therefore, the equation can be written as (x+4)(x-1)(x-a) = 0, where 'a' is the remaining solution. By setting the equation equal to zero and solving for 'a', we find that a = -3. So, the equation has three solutions: x = -4, x = 1, and x = -3.