Rational root theorem

The Rational root theorem (or rational zero theorem) is a proven idea in mathematics. It says that if the coefficients of a polynomial are integers, then one can find all of the possible rational roots by dividing each factor of the constant term by each factor of the leading coefficient. ^[1][2]

Think about this polynomial: $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0$

All of the possible rational roots are these: ±factors of a₀ / ±factors of a_n

This only finds the rational roots. There may be irrational or imaginary roots. Other theorems, such as Descartes’ rule of signs, help find how many imaginary roots there are for a given equation. ^[1]

Example

Think about this polynomial: 15x⁴ + 2x³ – 10x² + x – 8

Factors of the leading coefficient are: 15: ±1,±3,±5,±15

Factors of the constant are: -8: ±1,±2,±4,±8

Possible rational roots are: ±1/1, ±2/1, ±4/1, ±8/1, ±1/3, ±2/3, ±4/3, ±8/3, ±1/5, ±2/5, ±4/5, ±8/5, ±1/15, ±2/15, ±4/15, ±8/15

References

1. Larson, Ron. Algebra 2. Evanston, IL: McDougal Littell, 2007. Print.
2. Phillip S. Jones, Jack D. Bedient: The historical roots of elementary mathematics. Dover Courier Publications 1998, ISBN 0-486-25563-8, pp. 116–117