Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$.
I've been looking at some examples, but I get confused.
Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$.
I've been looking at some examples, but I get confused.
see http://en.wikipedia.org/wiki/Rational_root_theorem
Here $a_0=1$ and $a_n=1$
Factors of $a_0=1$ only and for $a_n$ too, it is $1$ only.
Thus, possible rational root is $\pm1/1=\pm1$
When we substitute $\pm 1$ back to the equation, they don't satisfy it $\implies \pm 1$(the only possible rational candidates) are not roots of the given equation $\implies $there is no rational root of $x^3-x+1=0$
For a better example, consider $6x^2-35$. The factors of $6$ are $\pm(1,2,3,6)$ and the factors of $35$ are $1,5,7,35$ (you only need the sign on one set). The possible rational roots are then $\pm(1,5,7,35,\frac 12, \frac 52, \frac 72 ,\frac {35}2,\frac 13, \frac 53, \frac 73 ,\frac {35}3,\frac 16, \frac 56, \frac 76 ,\frac {35}6)$. In this case, none works as the factorization is $(\sqrt 6x + \sqrt{35})(\sqrt 6x - \sqrt{35})$