# 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 an 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.

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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.
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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

StepEquationDescription
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.
7Simplify the exponents.
8Simplify the multiplication.
9Simplify 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

StepEquationDescription
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

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.

### 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.)