Associated Topics || Dr. Math Home || Search Dr. Math

### Rational Root Theorem

```
Date: 03/02/99 at 00:57:51
From: nicole
Subject: Rational Roots.

I do not know how to figure out a question like this:

Find all possible rational roots of:

4x^3 + 3x^2 + 6x + 10

I have no idea where even to start.

```

```
Date: 03/02/99 at 15:57:43
From: Doctor Rob
Subject: Re: Rational Roots.

The Rational Root Theorem says that if you have a polynomial with whole
number coefficients, like the above, and it has a rational root, then
the numerator of the root is a positive or negative divisor of the
constant term (10 in this case) and the denominator is a positive
divisor of the coefficient of the highest power term (4 in this case).
Then, the numerator can be 10, 5, 2, 1, -1, -2, -5, or -10, and the
denominator can be 1, 2, or 4. Furthermore, the numerator and
denominator would not share a common factor. That reduces the list of
possibilities to:

10/1, 5/1, 5/2, 5/4, 2/1, 1/1, 1/2, 1/4, -1/1, -1/2, -1/4, -2/1,
-5/1, -5/2, -5/4, -10/1.

These sixteen fractions (those with denominator 1 are actually whole
numbers) are the only possible rational numbers that could be roots of
the above equation. You can test them, one at a time. That will tell
you which are roots, if any.

By the way, it is not very hard to see that none of the positive
numbers can be roots because if you substitute them into the equation,
since all the coefficients are positive, the result will be a positive
number, and not zero. That will reduce your search to eight
possibilities.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Polynomials

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search