# How to convert these functions

• Feb 24th 2011, 11:19 AM
king.oslo
How to convert these functions
Hello there!

In my work, I will have to convert functions. My English is crap, so I will just show you what I mean.

If I have a function:

y = x + 1 i can convert it to look like this: x = y - 1. I would like to do the same for:

y = x^3 + x^2 + x + 1

or this:

y = x^4 + x^3 + x^2 + x + 1

or this:

y = x^5 + x^4 + x^3 + x^2 + x + 1

How is that done?

Thank you.

Kind regards,
Marius
• Feb 24th 2011, 11:38 AM
Ackbeet
For the first, you have this. For the second, you have this. For the third, in general, the quintic polynomial is not solvable (that's been proved, actually). Certain special cases can be solved, but not the general quintic.
• Feb 24th 2011, 12:17 PM
king.oslo
Thank you!

y = x^2 + x + 1

Kind regards,
Marius
• Feb 24th 2011, 12:22 PM
Ackbeet
That one's a quadratic. Rewrite as

$x^{2}+x+1-y=0,$ and use the quadratic formula to solve:

$x=\dfrac{-1\pm\sqrt{1-4(1-y)}}{2}=\dfrac{-1\pm\sqrt{4y-3}}{2}.$

Note that in order for everything to be real, you'd need $4y-3\ge 0,$ or $y\ge 3/4.$
• Feb 24th 2011, 12:52 PM
king.oslo
Thanks for that!

I found this one for the cubic:

But couldnt find one for the quartic on wikipedia. Do you know what it looks like?

Thanks! :) M
• Feb 24th 2011, 01:13 PM
Ackbeet
Quote:

Originally Posted by king.oslo
Hello there!

In my work, I will have to convert functions. My English is crap, so I will just show you what I mean.

If I have a function:

y = x + 1 i can convert it to look like this: x = y - 1. I would like to do the same for:

y = x^3 + x^2 + x + 1

This is a cubic.

Quote:

or this:

y = x^4 + x^3 + x^2 + x + 1
This is a quartic.

Quote:

or this:

y = x^5 + x^4 + x^3 + x^2 + x + 1
This is a quintic.

Quote:

How is that done?

Thank you.

Kind regards,
Marius
I don't know where you found your "cubic" formula, but it's not nearly complicated enough for a cubic. It looks like an incorrect quadratic. Here's the solution for your cubic. Here's the solution for your quartic.
• Feb 24th 2011, 01:33 PM
king.oslo
Quote:

Originally Posted by Ackbeet
This is a cubic.

This is a quartic.

This is a quintic.

I don't know where you found your "cubic" formula, but it's not nearly complicated enough for a cubic. It looks like an incorrect quadratic. Here's the solution for your cubic. Here's the solution for your quartic.

Thanks! :) M
• Feb 24th 2011, 02:47 PM
Ackbeet
You're welcome!
• Feb 25th 2011, 04:33 AM
HallsofIvy
A polynomial equation of degree 5 or higher cannot be solved in terms of radicals. There methods, using "special functions" for solving such equations.
• Feb 25th 2011, 04:47 AM
Ackbeet
Really? I didn't know that! Just doing a simple google search brought up the Jacobi theta function method, as well as hypergeometric functions (hypergeometric functions do seem to show up lots of different places, don't they? Maybe that's why the Russians do everything in terms of them.), as possibilities for solving the general quintic.