Factoring a difficult polynomial so that I can extend equation

[pugola12]

Homework Statement

Give a formula for the extended function that is continuous at the indicated point.

(x³-4x²-11x+30)/(x²-4), x=2

Homework Equations

The Attempt at a Solution

I know that I have to factor the top and bottom so that I can cancel terms that cause the function to be undefined at x=2, and I know how to do everything after that. However, I'm stuck on how to factor x³-4x²-11x+30. I know that the factors are (x-5)(x-2)(x+3), but I don't know how to go about getting there. I tried grouping, but it didn't work since there is no way that I see to factor -11x+30.
x³-4x²-11x+30

x²(x-4)-11x+30

I tried grouping other ways, but they did not work either.

 

[Dick]

In general, what you want is the rational roots theorem. If (x-r) is a factor of the numerator then r must divide 30. That gives a limited number of guesses for possible factors. But in this particular case, it's a pretty good guess that (x-2) is factor. Once you know that, divide the numerator by (x-2) to get a quadratic. Then factor the quadratic to get the other two factors.

 

[pugola12]

Oh gee, I completely forgot about that. (It's been a while since Algebra II) That makes total sense. Thank you so much for your help!