Ok, I said I was essentially done trying to factor quartics as the composition of quadratics. The one last question I had, even if I said I didn’t really have any, was if the following statement was true:

**Claim:** A quartic can be factored as the composition of two quadratics iff it is symmetric around .

What I have shown so far is that if it can be factored, then it is symmetric as claimed. I’ve shown that it can be factored iff . It remains to see that symmetry implies this relation on the coefficients.

Proof: If above is symmetric about , then the function is even. Plugging in to , expanding everything, and gathering up terms, one sees that

A polynomial is even if it only has even powers of . Since our is even, by our symmetry assumption, the linear coefficient above must be 0. If this happens, then it is easy to rearrange the equation and see that , as desired.

So there you have it. I’d summarize this work by saying that a quartic can be written as the composition of two quadratics iff it is symmetric about a vertical line.

### Like this:

Like Loading...

*Related*

Tags: polynomial, quartic

This entry was posted on November 18, 2008 at 12:20 am and is filed under Play. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

December 31, 2008 at 5:13 pm |

[…] should mention why I was thinking these curves, and why I want to draw that parametric curve. When I was playing with my quadric polynomials a while ago, I wanted to draw animations, but I had two parameters […]