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
a_{0}
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 2x^{3} +
5x^{2} - 4x - 3. Since
a_{0} = -3, the numerator of any
rational roots must be one of ±1, ±3.
Since a_{3} = 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 a_{n}. |
3 | |
Find all the factors of a_{0}. |
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 2x^{2} + 7x +
3. |
Example 1 |
Step | Equation | Description |
1 | |
This is the polynomial of which to find roots. |
2 | |
Find all the factors of a_{n}. |
3 | |
Find all the factors of a_{0}. |
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
- 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. https://www.allmathwords.org/en/r/rationalrootstheorem.html.
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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.)