Question

How do i know where to start when factoring?
x^3+x^2+9x+9

Answer 1

First, you should realize that ony nicely designed expressions will be factored just by looking at them. There is a general solution to a cubic equation and theoretically, this could be used to factorize such an expression. Trust me on this, you do NOT want to know THAT. For your expression, you just have to look for patterns.

x^3+x^2+9x+9

I find it suspicious that there is a '9' in both the last two terms. Let's use our distributive property of multiplication over addition and rewrite it a little:

x^3+x^2+9(x+1)

I see it now. I can do the same thing with x^2 in the first two terms.

(x^2)(x+1)+9(x+1)

Now we see a factgor of (x+1) in each of the two terms of the current expression. More Distributive Property

(x^2 + 9)(x+1)

I think we're done.

MOST importantly, I did not expect to see the entire solution when I started out. I just factored out some 9's. Then something else hit me.