Process for Finding Rational Zeroes
1. Utilizes the rational root theorem to list all possible rational zeroes of the polynomial P ( x )
2. Evaluate the polynomial at the numbers from the first step till we determine a zero. Let's imagine the zero is x = r , then we will know that it's a zero since P ( r ) =0 . Once it has been determined that it is actually a zero write the original polynomial as
P ( x )= ( x - r ) Q ( x )
3. Repeat the procedure using Q ( x ) this time rather than P ( x ) . This repeating will continue till we attain a second degree polynomial. At this instance we can directly solve this for the remaining zeroes.
To make simpler the second step we will utilizes synthetic division. This will very much simplify our life in various ways. First, remember again that the last number in the last row is the polynomial evaluated at r & if we do get a zero the remaining numbers in the last row are the coefficients for Q (x) and thus we won't ought to go back and determine that.
Also, in the evaluation step usually it is easiest to evaluate at the possible integer zeroes first and then go back and deal along with any fractions if we ought to.