Rational Roots Theorem
Pronunciation: /ˈræʃ nl rutz ˈθɪər əm/ ?
The
rational roots theorem
gives possible rational
roots
of a single variable
polynomial
with integer
coefficients.
A
rational roots 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 |
Cite this article as:
Rational Roots Theorem. 2009-02-06. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalrootstheorem.html.
Translations
Image Credits
Revision History
2009-02-06: Added vocabulary links (
McAdams, David.)
2009-01-29: Initial version (
McAdams, David.)