Find factors via rational root theorem

The rational root theorem is one of the most powerful, but least efficient, mechanisms for finding roots of a polynomial. The general rule of thumb is that the rational root theorem is the tool of last resort.

Rational Root Theorem: Derivation

As usual we will present the general case first, but follow it up with a specific concrete example so one can compare the two and see how the theorem works.

Ultimately our goal is to write a polynomial as a product of factors, something like (in the case of a factorable quadratic.) This form allows us to observe that the zeros of are the (rational) numbers and . Moreover, the expanded form of is (feel free to expand the factored form to verify this). Notice then that the two zeros are both of the form ”a factor of the constant term (of the expanded form) divided by a factor of the leading term (of the expanded form)” The rational root theorem aims to exploit this observation to generate a list of possible zeros of an unfactored polynomial.

Rational Root Theorem: Example

A concrete example should (hopefully) clarify how to use the rational root theorem in practice.

You can also watch a video of using the rational root theorem to fully factor a polynomial!