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Can you have a formula for every degree of polynomial?

  1. Feb 5, 2012 #1
    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. jcsd
  3. Feb 5, 2012 #2
    There is no general formula for polynomials of degree five or higher.
  4. Feb 5, 2012 #3
    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?
  5. Feb 5, 2012 #4
    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.
  6. Feb 5, 2012 #5
    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.
  7. Feb 5, 2012 #6
    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.
  8. Feb 5, 2012 #7
    What would they be expressible in then?
  9. Feb 5, 2012 #8


    Staff: Mentor

    There are a number of numerical methods that can give approximate solutions to polynomial equations.
  10. Feb 6, 2012 #9
    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.
  11. Feb 6, 2012 #10


    User Avatar
    Science Advisor

    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.
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