How can I factor a 3rd order polynomial using the cubic formula?

[Kiwiro0ls]

Homework Statement

What are the steps to factoring 3rd order polynomials like x^3+8x^2-21x+10?
It's to find eigenvalues of a matrix in linear algebra, I completely forgot how to factor and it's killing me.

Homework Equations

The Attempt at a Solution

None, unless its a polynomial that I can factor by grouping, I have no clue how to begin.

 

[Mute]

EDIT: Nevermind the below, that factor trick (the 'rational root test') doesn't work in this case. The discriminant is positive, but it turns out that all three roots are irrational.

For a cubic polynomial with integer coefficients, which is what you have, I believe one of the roots is typically a factor of the constant term. So, try the factors of 10 and see if one of them is a root. Once you've figured out one of the roots, you can factor an (x-root) term out by polynomial division, leaving you with (x-root)*quadratic, and the quadratic you can factor with the quadratic formula.

 

[Kiwiro0ls]

:( so the pq thing with synthetic and what-the-other-one's-name division?
I don't know why i didnt think of it (but that sounds really tedious).
Thanks for your help! :)

 

[Kiwiro0ls]

Got it, so i happened to pick the only example in the book of a matrix with irrational eigenvalues. I'm still not so sure how to find the irrational roots of a 3rd order polynomial... i should change the name of this thread haha.

 

[eumyang]

There's always the cubic formula, but using it is a bit... tedious.

 

[DuncanM]

There's always the cubic formula, but using it is a bit... tedious.

To add to what "eumyang"" said, using the Cubic Formula to get an exact algebraic result is, indeed, a bit tedious.

On the other hand, if you are fine with numerical results, there are several free online utilities available; some of them are listed at the bottom of that Wikipedia page.

In addition, I refer you to an old post I made at the bottom of the following thread:

https://www.physicsforums.com/showthread.php?t=580075&page=2

A search of these forums would probably turn up much more information about factoring a cubic polynomial.