Linear (1st power), Quadratic (2nd power) what's Next?

[bballwaterboy]

We learned in class that linear equations have variables that are raised to a highest power of one (x) and that quadratic equations have variables raised to a highest power of two (x^2). What happens when you get equations with variables raised to powers of three (x^3)...four (x^4), etc. etc. etc.?

Are there names for these and do they have special rules for solving them too? Just very curious. :)

 

[pwsnafu]

Cubic, Quartic, Quintic.
Cubic and quartic formulas do exist. It is not possible to express similar formulas for quintics and higher.

 

[bballwaterboy]

Very cool. Out of curiosity, why are quadratics called quadratic?

"Qua" or "Quad" doesn't express the value of two normally, right? Wouldn't "bi" be a better type of beginning to the word? :p

 

[jbriggs444]

Supposedly it traces back to the Latin word "quadratus" meaning "square". A quadratic equation involves a squared term, of course.

 

[AMenendez]

As mentioned above, they have names (cubic ($ax^3+\dots$), quartic ($ax^4+ \dots$), etc.). As for solving them, it can be done a number of ways. Factoring out powers of $x$ works in most cases; so, if you had a cubic like $x^3 + 18x^2 + 81x$, you could factor out a power of $x$ and get $x(x^2 + 18x + 81)$, where you get a quadratic inside the parentheses, which can be factored. Hence, you get $x(x+9)(x+9)$, with solutions $x = 0$ and $x = -9$. You can decompose higher order functions like quartics using the same principles, but doing so you usually end up making substitutions, so it gets more complicated.

 

[bballwaterboy]

Supposedly it traces back to the Latin word "quadratus" meaning "square". A quadratic equation involves a squared term, of course.

That's pretty funny! Square...squared. heh heh :w

 

[bballwaterboy]

Cubic, Quartic, Quintic.
Cubic and quartic formulas do exist. It is not possible to express similar formulas for quintics and higher.

Very cool, pwsnafu. Any reason behind the inability to express formulas for higher powers?

 

[pwsnafu]

Very cool, pwsnafu. Any reason behind the inability to express formulas for higher powers?

The result is known as the Abel-Ruffini theoerm. And yes there is indeed a very deep reason using Galois theory. But I cannot explain at a level you would be able to understand. Maybe someone else can try.

 

[bballwaterboy]

The result is known as the Abel-Ruffini theoerm. And yes there is indeed a very deep reason using Galois theory. But I cannot explain at a level you would be able to understand. Maybe someone else can try.

Don't even worry about it! Those already sound way over my head. :w But very cool to know still. THanks.

 

[statdad]

"Factoring out powers of x works in most cases;"

Most cases? Not at all.

 

[epenguin]

Very cool, pwsnafu. Any reason behind the inability to express formulas for higher powers?

The human story is one of the most oft-told tales in maths, subject of many articles and books. Most oft told no doubt by Iain Stewart, also by ET Bell, recently in the book "The Equation that couldn't be solved" by Mario Livio, in one or two old articles in Scientific American and in almost every pop math book - I've read many - though they give the background but when they get to the real proof argument they start hand waving and I have never got to the real algebra.

What "algebraic unsolvability" means is that, while for the quadratic you can write a so-called solution you know x = [- b ± √Δ]/2a where Δ = b² - 4ac whereas for degree >4 you cannot write a solution x = any formula like that in terms of n-th roots of expressions in the coefficients.

But what does the above "solution" of the quadratic mean? It means from the coefficients you can straightforwardly calculate a number Δ from which you can less straightforwardly calculate another number √Δ meaning a number which when squared is Δ. Except in general you can't exactly. You can just calculate a number which squared is close to Δ. Starting with that one number you can calculate another number (e.g.to an extra decimal point) which squared is closer to Δ and then again and again, closer and closer but never getting to something whose square is exactly Δ. And how'do you do even that you may or may not remember or be able to work out - let's say we've swept a bit under the carpet when we write √Δ.

To solve a general quintic or higher equation you have to start with a set of numbers and perform a not dissimilar series of operations to get closer and closer to a number which when substituted in the polynomial gives you zero.

So this huge and trumpeted difference between 'algebraic' solution on the one hand and 'numerical' solution is really the difference between one infinite process that can start with one number and give you one closer and closer to 0, and a slightly different process that uses several numbers to do the same thing. In both cases solving f(x) = 0 means finding a series of numbers x₀, then x₁ then ... In which f(x_r) gets closer and closer (efficiently) to 0 as r increases.

Algebraic solvability is a question of mathematical structure. Nontrivial, deep and difficult no doubt. It has nothing to do with practical numerical solvability (I.e. approximation) which can 'always' be done (as a matter of fact I believe people who really have to solve a lot of cubic or quartic equations for practical purposes do not bother to use the algebraic method).

Subject to correction, I prefer to say refinement :D, by the mathematicians.