The discussion revolves around the existence of formulas for solving polynomial equations of various degrees, particularly focusing on the implications of the Abel-Ruffini theorem regarding polynomials of degree five and higher. Participants explore whether it is possible to generalize solutions for all polynomial degrees or if individual formulas can be constructed for specific cases.
Participants generally disagree on the possibility of having a general formula for all polynomial degrees, with some asserting that such formulas do not exist beyond degree four, while others explore the potential for individual solutions. The discussion remains unresolved regarding the implications of the Abel-Ruffini theorem and the nature of solutions for higher-degree polynomials.
Participants express uncertainty about the specific limitations of the Abel-Ruffini theorem and the conditions under which higher-degree polynomial solutions can be expressed. There is also a lack of consensus on the methods available for solving these equations and the nature of the solutions.
Jorriss said:There is no general formula for polynomials of degree five or higher.
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 said: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 said: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.
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 said: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 said: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.
There are a number of numerical methods that can give approximate solutions to polynomial equations.questionpost said:Well how am I suppose to solve a higher degree polynomial that I can't factor?
questionpost said: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.
Mark44 said:There are a number of numerical methods that can give approximate solutions to polynomial equations.