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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.

Please help!


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

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