Final answer:
To find the roots of the cubic polynomial p(x)=x³-3x²-4x+12, we can use methods like factoring by grouping, synthetic division, or the Rational Root Theorem. Once a root is found, the polynomial can be reduced to a quadratic equation, and the remaining roots can be found.
Step-by-step explanation:
Finding the Roots of a Cubic Polynomial
To find the roots (zeros) of the polynomial p(x)=x³-3x²-4x+12, we can use factoring by grouping, synthetic division, or numerical methods if the factors are not readily apparent. However, unlike quadratic equations of the form ax²+bx+c = 0, for which the quadratic formula can be used, cubic equations do not have such a straightforward formula for finding roots.
To find the roots of the polynomial, we may first check whether the polynomial has any rational roots using the Rational Root Theorem. Any rational root should be a factor of the constant term (12) divided by a factor of the leading coefficient (1 in this case). Once a single root is found, we can use synthetic division to reduce the polynomial to a quadratic of the form ax²+bx+c=0, and then solve for the remaining roots.
If necessary, numerical methods such as the Newton-Raphson method can also be applied to find approximations of the roots when they cannot be expressed in radical form.