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I was reading about quintic equations and this came up:

In the early nineteenth century, Paolo Ruffini (1765–1822) and Niels Henrik Abel (1802–1829) proved that no such general formulas utilizing the usual operations and root operations exist. This means that there will never be a simple formula that provides the solutions for every single a, b, c, d, e, and f in a quintic equation.

so, I am a little confused. isn't the goal of solving a quintic equation is to find X? what is the role of coefficients in here? can anyone explain?

Daruis soli
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  • If you have no coefficients, do you even have a quintic to solve for $x$? – David K Jun 11 '19 at 16:50
  • Not sure what you are asking. It is, of course, true that $\textit {some}$ quintics are solvable. $x^5$ for example, or $(x+1)(x+2)(x+3)(x+4)(x+5)$. The Theorem says that there isn't a "sensible" formula, akin to the quadratic formula, that solves all quintics regardless of the coefficients. – lulu Jun 11 '19 at 16:59

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The role of coefficients is to specify which polynomial function is in the problem. There is no general solution in radicals analogous to, say, the quadratic formula; that's what the theorem is saying. (In the quadratic case, the formula writes roots in terms of coefficients.)

J.G.
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  • thank you very much. I did some digging and still have problems in understanding this part of the above context: "the solutions for every single a, b, c, d, e, and f in a quintic equation." What is this mean? is the writer's intention is that for every arbitrary a, b, c, d, e, and f in a quintic equation? – Daruis soli Jun 11 '19 at 20:33
  • @Daruissoli In other words, solving any problem of the form $ax^5+bx^4+cx^3+dx^2+ex+f=0$. – J.G. Jun 11 '19 at 20:41