In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x^3−2x^2−11x+12

The Rational Root Theorem provides a method for identifying all possible rational zeros of a polynomial function. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem helps narrow down the candidates for testing as potential zeros.

Synthetic division is a simplified form of polynomial long division that allows for efficient division of a polynomial by a linear factor. It involves using the coefficients of the polynomial and the potential zero to perform the division, yielding a quotient and a remainder. If the remainder is zero, the tested value is indeed a zero of the polynomial.

Once an actual zero is found using synthetic division, the polynomial can be expressed as a product of the linear factor corresponding to the zero and a reduced polynomial. The remaining zeros can then be found by factoring or using the quadratic formula on the resulting polynomial, allowing for a complete solution to the original polynomial equation.