# 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 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:

McAdams, David E. Rational Roots Theorem. 5/5/2011. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalrootstheorem.html.
### Image Credits

### Revision History

2/6/2009: Added vocabulary links. (

McAdams, David E.)
1/29/2009: Initial version. (

McAdams, David E.)