Can you have a formula for every degree of polynomial?

[questionpost]

I have some math people who say you can, and some who say you can't beyond quintic because of the Abel-Ruffini theorem. Which is it? Can I generalize all polynomials? Or at least can I manually make a formula for each individual degree?

 

[Jorriss]

There is no general formula for polynomials of degree five or higher.

 

[questionpost]

There is no general formula for polynomials of degree five or higher.

So even if I have it set to 0, there's no possible way to get a single formula for a hex-tic polynomial? What if I write it in the formula to break it down into 3 different quadratic equations?

 

[Jorriss]

So even if I have it set to 0, there's no possible way to get a single formula for a hex-tic polynomial? What if I write it in the formula to break it down into 3 different quadratic equations?

Of course you can come up with examples of polynomials that are easily factorizable, but there is no general formula for, say, ax^6 + bx^5 + cx^4 + dx^3 + fx^2 + gx^1 + h = 0.

 

[questionpost]

Of course you can come up with examples of polynomials that are easily factorizable, but there is no general formula for, say, ax^6 + bx^5 + cx^4 + dx^3 + fx^2 + gx^1 + h = 0.

Well how am I suppose to solve a higher degree polynomial that I can't factor? Also, I know the abel-ruffini theorem exists, but I don't get exactly why it says you can't have formulas bigger than 5th degree.

 

[lugita15]

Well how am I suppose to solve a higher degree polynomial that I can't factor? Also, I know the abel-ruffini theorem exists, but I don't get exactly why it says you can't have formulas bigger than 5th degree.

You can have formulas, but those formulas would not be expressible in terms of elementary algebraic operations, specifically addition, subtraction, multiplication, division, and taking roots.

 

[questionpost]

You can have formulas, but those formulas would not be expressible in terms of elementary algebraic operations, specifically addition, subtraction, multiplication, division, and taking roots.

What would they be expressible in then?

 

[Mark44]

Well how am I suppose to solve a higher degree polynomial that I can't factor?

There are a number of numerical methods that can give approximate solutions to polynomial equations.

Also, I know the abel-ruffini theorem exists, but I don't get exactly why it says you can't have formulas bigger than 5th degree.

 

[questionpost]

There are a number of numerical methods that can give approximate solutions to polynomial equations.

Why not an exact answer? If there's a specific process being done to the input, how is there not a specific answer? That doesn't even make sense. You don't type in y or z ≈x, you type in y or z = x. What about logs? that's not a subtraction or addition or multiplication or division or root, what about exponents? or vectors?
I just don't get how I could input a number in x^7+3x^3-12 and get an exact answer but if I work backwards I somehow don't get that exact input I started with.

 

[HallsofIvy]

The point of Abel-Ruffini is that, for n greater than 4, there exist polynomials of degree n having zeroes that cannot be written in terms of radicals. There can exist formulas for roots of such polynomials that do not use radicals. I believe, but am not certain, that there is no one formula for all n.