Final answer:
To solve the cubic equation f(x) = x^3 - 4x^2 - 16x + 8 in the interval [-4, 4], we can use the Rational Root Theorem to find possible rational roots. By testing the possible rational roots, we find that -2 is a solution. Dividing f(x) by (x+2), we obtain a quadratic factor x^2 - 6x + 4, which can be solved to find the roots 3.27 and 2.73.
Step-by-step explanation:
The given expression is a cubic equation of the form f(x) = x^3 - 4x^2 - 16x + 8. To solve this equation, we can use the Rational Root Theorem to find possible rational roots. The possible rational roots can be determined by listing all the factors of the last term (8) and dividing them by the factors of the leading coefficient (1). After trying the possible rational roots, we find that the root -2 is a solution. Using synthetic division or long division, we can divide f(x) by (x+2) to obtain the quadratic factor x^2 - 6x + 4. We can then solve this quadratic factor using the quadratic formula or factoring. The roots of this quadratic factor are approximately 3.27 and 2.73. Therefore, the solutions to the original equation f(x) = x^3 - 4x^2 - 16x + 8 are -2, 3.27, and 2.73 in the interval [-4, 4].