# Thread: Find a Fixed Cubic

1. ## Find a Fixed Cubic

I am working on a problem, the remaining step I need to do is very computational.
I am mathematically software challenged, can some one find the solution to this problem:

Find $a,b,c,d,e,f,g,h$ (real numbers) and with $a,e\not = 0$ so that if in the function $f(x) = \tfrac{ax^3+bx^2+cx+d}{ex^3+fx^2+gx+h}$ we replace $x$ by $\tfrac{ - x - \sqrt{3}}{\sqrt{3}x-1}$ then we end up with the same function, i.e. $f(x) = f\left( \tfrac{-x-\sqrt{3}}{\sqrt{3}x-1} \right)$.

2. Take a look at this. I think I understand your question. I'm holding x as 1 for both sides. I take the value of the 2nd function using x=1, and then plug that in as the x value to the main F(x). Do you want me to write something to loop through a-h to find your target values?

3. Thank you, but like I said I am mathematically software challenged I do not even have Excel.

4. Originally Posted by ThePerfectHacker
Thank you, but like I said I am mathematically software challenged I do not even have Excel.
It says
a=-1
b=-20
c=-30
d=-243
e=-2
f=6
h=8

5. Originally Posted by Mathstud28
It says
a=-1
b=-20
c=-30
d=-243
e=-2
f=6
h=8
Okay I will try that. Can you check that, if you have computer-algebra system on your computer. I am too lazy to.

6. Originally Posted by ThePerfectHacker
Okay I will try that. Can you check that, if you have computer-algebra system on your computer. I am too lazy to.
I don't, I'm sorry

7. I just tried graphing this function and it does not work.
I used $g=0$, because you never told me what value is, I assumed it was zero.

8. Sorry gentleman, I should have described better. The numbers on that file are starting points, not the answer.

My question to you was do you want me to set x = 1 for both equations. After that, my thought was run my program to come up with a combination for each to match equations.

If that's what you want, I'll program that.

9. Originally Posted by mathceleb
If that's what you want, I'll program that.
I am looking for coefficients that when I substitute the linear rational function and after simplifying denominator and all that stuff I end up with the same expression as I started. So it seems this is a linear algebra problem with 8 variables.

10. Originally Posted by ThePerfectHacker
I am looking for coefficients that when I substitute the linear rational function and after simplifying denominator and all that stuff I end up with the same expression as I started. So it seems this is a linear algebra problem with 8 variables.
Ok, that makes sense. Let me take a look and see if I can code this for you. My "day job" is really busy this week, so I'll see what I can do this week after hours.

11. Originally Posted by ThePerfectHacker
Thank you, but like I said I am mathematically software challenged I do not even have Excel.