Rational Root TheoremDate: 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/ |
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