Finding an Orthogonal Polynomial to x^2-1/2 on L2[0,1]

[mandygirl22]

Find a polynomial that is orthogonal to f(x)=x²-1/2 using L²[0,1].

I have looked all in the textbook and all over the internet and have found some hints if the interval is [-1,1], but still do not even know where to start here. I think I was gone the day our professor taught this because I do not know anything about it and the book does not make any sense out of it. Thanks for your help!

 

[HallsofIvy]

Well, just DO it! The simplest kind of polynomial is linear. Integrate Ax+ B times [math]x^2- 1/2[/math] from 0 to 1 and choose A and B so the integral is 0.

 

[mandygirl22]

*hits forehead* I knew there had to be some simple way of doing it that I was ignoring! Thanks!

(and the REALLY HARD solution ended up being f(x)=x)