# Rational Roots of a Polynomial

Rational Roots of a Polynomial

Let be a polynomial with integer coefficients and constant coefficient ≠0. Use this Demonstration to find the rational roots of .

P(x)=++⋯+x+

n

x

a

n-1

n-1

x

a

1

a

0

a

0

P(x)

Each rational root is of the form , where and are integers such that divides and divides , the leading term. Make a list of all the possible rational roots by considering divisors of and .

±/

k

0

k

n

k

0

k

n

k

0

a

0

k

n

a

n

a

0

a

n

At the start, the set of rational roots found is empty. Choose a candidate from the list. Using the Ruffini–Horner algorithm, divide by to get a polynomial and remainder (cyan box). If , then , and is a root of ; add to . Repeat this process with and the next candidate; continue until all the rational roots have been found. (The maximum number of roots is , so there may be no need to test all the candidates.)

S

h

P(x)

x-h

Q(x)

R

R=0

P(x)=Q(x)(x-h)

h

P(x)

h

S

P(x)/(x-h)

n

When =1, the rational roots are integers.

a

n