Polynomial equations with real coefficients can have three types of roots:  rational, irrational, complex. 

The Rational Root Theorem

Suppose that a polynomial equation with integral coefficients has the root h/k, where h and k are relatively prime integers.  Then h must be a factor of the constant term of the polynomial and k must be a factor of the leading coefficient. 

To find rational roots: 

1)  List all +/- factors of the constant (h) and the leading coefficient (k). 

2)  List all possibilities of rational roots:  +/- h/k. 

3)  Use Descartes’ rule of signs to figure out how many positive, negative, and complex roots there might be.  Eliminate some of the possible roots if possible. 

4)  Put the remaining possible roots in order from smallest to largest. 

5)  Choose a possible rational root that falls in the middle of the pack.  Test it using synthetic division.  If you get a remainder of 0, it’s a root!  Find the depressed equation, and look for roots of the depressed equation. 

6)  After doing synthetic division with a positive divisor, if a number is not a root, but you get all positives, then the number is too high (upper bound).  Disregard all higher possibilities and look lower. 

7)  After doing synthetic division with a negative divisor, if a number is not a root, but the signs of the answer are alternating positive and negative, then the number is too low (lower bound).  Disregard all lower numbers and try higher.   

8)  If you test several possible rational roots, nothing works, you find the upper and lower bounds but no roots, then the roots are irrational.  There are no rational roots.

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