How can I factor a cubic equation by hand to find the x-intercepts?

[DecayProduct]

This is not a homework question, per se, because I'm not a student. But it is a problem I found in a book. Actually, the problem doesn't involve what I'm going to ask, but it did present an opportunity for me to explore the subject.

I have the graph of $$f(x)=x^{3}+3x^{2}-9x+3$$. I know the x intercepts of the function from looking at the graph, what I want to know is how to factor the equation by hand to derive those intercepts.

$$x(x^{2}+3x^{2}-9)=-3$$ doesn't help because factoring the quadratic in the middle gives me the zeros of that particular piece, which are meaningless, because I'm not looking for those zeros. I've tried factoring by pieces, but I can't get the right products to pop out.

Surely, because the graph exists at all, and because the function is continuous across the x-axis, then there must be a way to factor the zeros out of $$f(x)$$? Or am I totally off base with that assumption?

 

[slider142]

If you know that the number a is a root of the equation, then it is trivial to factor out the binomial x - a from the cubic polynomial. Otherwise, you can only really get as far as the rational roots test before having to resort to iteration algorithms to converge to irrational roots (ie., the Newton-Raphson method).

 

[Redbelly98]

If the http://en.wikipedia.org/wiki/Rational_root_theorem" without resorting to numerical approximations. (See the two clickable links in that previous sentence for more details.)

p.s...

Moderator's note:

Any textbook-style exercises are to be posted in the appropriate forum in our https://www.physicsforums.com/forumdisplay.php?f=152" area. This should be done even if the problem is part of one's independent study. Basically, it's prohibitively difficult to moderate a lot of questions based on whether it's for an actual course assignment.

I have gone ahead and moved this thread to our "Homework" forums.

 

[fedderenator]

Try synthetic division...