Solving Polynomials: Factoring x^3-2x^2-x+2

[mbrmbrg]

I'm going over some math work, and ran across the following:
$$\frac{2x^2-5x-1}{x^3-2x^2-x+2}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{x-2}$$

How do you get $$x^3-2x-x+2=(x-1)(x+1)(x-2)$$?

 

[Hurkyl]

Well, it's easy to verify that it's right.

To find it... well, one way is that there are only four rational numbers that could possibly be roots of your polynomial, so you just try them!

 

[mbrmbrg]

To find it... well, one way is that there are only four rational numbers that could possibly be roots of your polynomial

How so? (pardon me if I be asking heap stupid question; I pulled an all-nighter last night and the brain doth rebel against unwarranted abuse)

 

[Hurkyl]

http://en.wikipedia.org/wiki/Rational_root_theorem

 

[mbrmbrg]

Why do I feel like someone at PhysicsForums sent me to this webpage before...?

Thanks!

 

[Tedjn]

Sometimes, especially for third degree expressions, you can see if you are able to factor by grouping. Namely,

$$x^3 - 2x^2 - x + 2 = (x^3 - 2x^2) + (-x + 2) = x^2(x - 2) - (x - 2) = (x - 2)(x^2 - 1) = (x - 2)(x - 1)(x + 1)$$