Rational Roots Theorem

Pronunciation: /ˈræʃ.nl rutz ˈθɪər.əm/ Explain

The rational roots theorem gives possible rational roots of a single variable polynomial with integer coefficients. A rational root is a root of a polynomial that is a rational number. Given a polynomial any rational roots of the polynomial have a factor of a₀ as a numerator and a factor of a_n as a denominator. The rational roots theorem is also called the rational zeros theorem.

Start with the polynomial 2x³ + 5x² - 4x - 3. Since a₀ = -3, the numerator of any rational roots must be one of ±1, ±3. Since a₃ = 2, the denominator of any rational roots must be one of ±1, ±2.

To see why, start with the two factors Setting each factor to 0 gives the roots of the polynomial: and The roots of the polynomial ( 3x + 2 )( 5x - 7 ) are x = 2/3 and x = -7/5.

Now multiply the two factors of the polynomial. According to the Rational Roots Theorem, any rational roots of the polynomial will be a factor of -14 divided by a factor of 15. The factors of -14 are ±1, ±2, ±7, ±14. The factors of 15 are ±1, ±3, ±5, ±15.

Examples

Step	Equation	Description
1		This is the polynomial of which to find roots.
2		Find all the factors of an.
3		Find all the factors of a0.
4		Calculate all the possible rational roots by dividing the factors of -3 by the factors of 2.
5		Simplify any fractions that can be simplified.
6		Test the root x=1 by substituting 1 in for x.
7		Simplify the exponents.
8		Simplify the multiplication.
9		Simplify the addition. Since 0 = 0 is a true statement, 1 is a root of P( x ).
10		Use synthetic division to find the remaining factor.
11		Here are the factors of the polynomial. Use the quadratic equation to find any roots of the quadratic 2x2 + 7x + 3.
Example 1

Step	Equation	Description
1		This is the polynomial of which to find roots.
2		Find all the factors of an.
3		Find all the factors of a0.
4		Calculate all the possible rational roots by dividing the factors of -3 by the factors of 2.
5		Simplify any fractions that can be simplified.
6		Test the root x = 3 by substituting 3 in for x.
7		Simplify the exponents.
8		Simplify the multiplication.
9		Simplify the addition and subtraction. Since 84 ≠ 0, 3 is not a root of P(x).
Example 2

References

1. McAdams, David E.. All Math Words Dictionary, rational roots theorem. 2nd Classroom edition 20150108-4799968. pg 151. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Rational Roots Theorem. 5/2/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalrootstheorem.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

5/2/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/4/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
2/6/2009: Added vocabulary links. (McAdams, David E.)
1/29/2009: Initial version. (McAdams, David E.)