Rational root theorem
In algebra, the rational root theorem states that given an integer polynomial with leading coefficent
and constant term
, if
has a rational root in lowest terms
,
and
.
This theorem aids significantly at finding the "nice" roots of a given polynomial, since the coefficients entail only a finite amount of rational numbers to check as roots.
Proof
Let be a rational root of
, where all
are integers; we wish to show that
and
. Since
is a root of
,
Multiplying by
yields
Using modular arithmetic modulo
, we have
. As
and
are relatively prime,
. Via similar logic in modulo
,
, which completes the proof.
Problems
Here are some problems that are cracked by the rational root theorem. The answers can be found here.
Introductory
- Factor the polynomial
.
Intermediate
- Find all rational roots of the polynomial
.
- Prove that
is irrational, using the Rational Root Theorem.
Answers
1.
2.
3. A polynomial with integer coefficients and has a root as
must also have
as a root. The simplest polynomial is
which is
. We see that the only possible rational roots are
and
, and when substituted, none of these roots work.