# Can you have a formula for every degree of polynomial?

1. Feb 5, 2012

### 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?

2. Feb 5, 2012

### Jorriss

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

3. Feb 5, 2012

### questionpost

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?

4. Feb 5, 2012

### Jorriss

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.

5. Feb 5, 2012

### questionpost

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.

6. Feb 5, 2012

### lugita15

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.

7. Feb 5, 2012

### questionpost

What would they be expressible in then?

8. Feb 5, 2012

### Staff: Mentor

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

9. Feb 6, 2012

### questionpost

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.

10. Feb 6, 2012

### HallsofIvy

Staff Emeritus
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.