In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x^3−10x−12=0

The Rational Root Theorem states that any rational solution of a polynomial equation, in the form of p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps in identifying all possible rational roots of a polynomial, which can then be tested to find actual roots.

Synthetic division is a simplified form of polynomial long division that allows for the division of a polynomial by a linear factor. It is particularly useful for finding the quotient and remainder when a polynomial is divided by a binomial of the form (x - r), where r is a root. This method streamlines the process of finding remaining roots after identifying one root.

Factoring polynomials involves expressing a polynomial as a product of its factors, which can be linear or quadratic expressions. Once a polynomial is factored, it becomes easier to solve for its roots by setting each factor equal to zero. This concept is essential for simplifying polynomial equations and finding all possible solutions.