Rational roots

Sep 2015
Los Angeles
Hey guys! I'm in pre calculus not sure if this is the right forum but my problem says "list the possible rational roots of the equation. Then determine the rational roots" and the equation is x(to the 3rd) + 2x(to the 2nd) -6x +3=0 been working on it for an hour and am lost. Appreciate the help thanks!!


Matt Westwood

MHF Hall of Honor
Jul 2008
Reading, UK
It should be straightforward to factorise. Try x+3 and if that doesn't work, try x-3. If neither of those work then try x+1 and x-1. Something should give sooner or later.


MHF Helper
Jun 2008
North Texas
What does the rational root theorem say about the form of possible rational roots? ... does something about the ratio of factors of the constant term and leading coefficient ring a bell?
Sep 2015
Everything is so much easier with calculus. Don't let anybody convince you otherwise; odds are anybody who would make such a claim to the contrary has never studied calculus.
I did a precalculus textbook over a three month period, and I was just as confused as you were when it came to finding roots not easily discoverable using factorization and the Quadratic formula. Now I'm teaching myself calculus. Chapter 3 taught an amazingly simple and powerful method of finding roots of a function, which I share with you below:

Newton's Mathod.jpg

It's called Newton's Method, and all you need is elementary differential calculus. Still probably a distant future for you, but one that, with the right motivation, you will be able to make for yourself with effort and perseverance.


MHF Helper
Apr 2005
Since this problem specifically asks about rational roots, you are clearly expected to know, and use, the "rational root theorem", that skeeter referred to. It is this:
"if the polynomial equation \(\displaystyle a_nx^n+ a_{n-1}x^{n-1}+ \cdot\cdot\cdot+ a_1x+ a_0= 0\), with integer coefficients, has the rartional number \(\displaystyle \frac{p}{q}\) as a solution, then the denominator, q, must evenly divide the leading coefficient, \(\displaystyle a_n\), and the numerator, p, must evenly divide the constant term, \(\displaystyle a_0\).

In this problem the leading coefficient is 1 so the denominator of any rational root must evenly divide 1 which means it must itself be 1 or -1 which means any rational root must be an integer. The constant term is 3 which has, as factors, only 1, -1, 3, or -3.

Zaqiqu, you understand that "Newton's method" only gives a numerical approximation, not an exact solution, and so is not appropriate for this problem, don't you?