Final answer:
The equation x³+3x²-6x-8=0 is solved by using the Rational Root Theorem to identify possible rational roots and then testing them through synthetic division or substitution. Once a root is found, the polynomial is divided by (x-r) to reduce it to a quadratic equation, which can be solved using the quadratic formula.
Step-by-step explanation:
You asked how to solve the polynomial equation x³+3x²-6x-8=0 using the Rational Root Theorem. The Rational Root Theorem states that any rational solution, when written in lowest terms p/q, p must be a factor of the constant term and q must be a factor of the leading coefficient. For the given polynomial, the constant term is -8 and the leading coefficient is 1, which means the possible rational roots are ± 1, ± 2, ± 4, and ± 8. We can test these potential roots by either synthetic division or direct substitution.
If we find a root r, we can then divide the polynomial by (x-r) to get a quadratic equation of the form ax²+bx+c=0, which can be solved using the quadratic formula.