Final answer:
To find the remaining roots of the polynomial function given that (x + 2) is a factor, divide the polynomial by (x + 2) to get a cubic polynomial. Then, use methods such as synthetic division, the quadratic formula, or numerical methods to solve for the rest of the roots.
Step-by-step explanation:
The question asks to find the remaining roots of a polynomial function f(x) = x⁴ - 4x³ - x² + 16x - 12, given that (x + 2) is one of its factors. To find the remaining roots, we first perform polynomial division by dividing f(x) by (x + 2) to obtain the reduced polynomial. After the division, we now have a cubic polynomial that we then need to factor further or solve for roots using methods such as synthetic division, factoring by grouping, or applying the quadratic formula if it reduces to a quadratic equation.
Once we have the reduced polynomial, we look for rational roots using the rational root theorem or numerically via graphing methods or iterative methods like Newton's method if necessary. If any complex roots exist, they usually come in conjugate pairs. The roots of the cubic polynomial are the remaining roots of f(x) once we have accounted for the factor (x + 2).
To aid in solving such a cubic polynomial and subsequent quadratics that may result, one can use the quadratic formula ax²+bx+c = 0 with the following formula to solve for x:
√(b² - 4ac) / 2a.