#### Alfredo Dawlabany

##### New member

- Joined
- Aug 22, 2017

- Messages
- 45

The question is that the second method for factoring I mentioned is not always valid. Why ? And when it is valid ?

And also is there a proof for it ?

- Thread starter Alfredo Dawlabany
- Start date

- Joined
- Aug 22, 2017

- Messages
- 45

The question is that the second method for factoring I mentioned is not always valid. Why ? And when it is valid ?

And also is there a proof for it ?

- Joined
- Jun 18, 2007

- Messages
- 18,149

Go to:

The question is that the second method for factoring I mentioned is not always valid. Why ? And when it is valid ?

And also is there a proof for it ?

https://www.google.com/search?q=rational+root+theorem&rlz=1C1GGRV_enUS748US748&oq=rational+root+theorem&aqs=chrome..69i57j0l5.8933j0j8&sourceid=chrome&ie=UTF-8

and read some of the sites mentioned there.

To me, this sounds like you're talking about theAnd also I know that one of the divisors of the constant term of a polynomial is a root of it.

The question is that the second method for factoring I mentioned is not always valid. Why ? And when it is valid ?

As an example, take the polynomial \(\displaystyle 2x^3+3x^2+4x+16\). If we apply the rational root theorem, we see that the possible rational roots are \(\displaystyle \pm 1, \pm 2, \pm 4, \pm 8, \pm 16, \text{ and } \pm \dfrac{1}{2}\). However, checking these roots reveals that none of them are actually roots. In fact, the one real root is at \(\displaystyle x \approx 2.2211\) and the other two roots are complex.

The rational root theorem gives you the list of every RATIONAL number that MAY BE a zero of a polynomial in one unknown with INTEGER coefficients. It does not say that there is such a zero.

A special version applies if the coefficient of the highest order term is 1. Then you can apply the integer root theorem, which gives you the list of every INTEGER that MAY BE a zero of a polynomial in one unknown with INTEGER coefficients. Again, it does not say that there is such a zero.

For an in depth discussion, start with https://en.m.wikipedia.org/wiki/Rational_root_theorem