Quartic and quintic equations?

Question

Assuming that the equation is not factorizable, is there a formula to calculate all the roots of a 4th and 5th polynomial equation given the coefficients of powers of x?

ax^4 + bx^3 + cx^2 + dx + e = 0
ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0

Thanks.

If there is a formula to solve quartic equations, what is it?

Answer 1

Yes for quartic, but it is very complicated and takes a lot of time to solve it.
Formula: http://en.wikipedia.org/wiki/Quartic_equation

However, there is no formula for quintic or higher (6 and above) degree polynomials, as proven in 1820.

Answer 2

quartic yes, but quintic no. The method for quartic equations is quite complicated, and merely reduces the quartic to an equivalent cubic, which you presumably already know how to solve. It's usually not taught because it is so complicated that no one would ever use it in a practical sense. It's mostly or theoretical interest. If you want the rundown, look at Ferrari's solution at http://en.wikipedia.org/wiki/Quartic_equation. Fermat also provided a solution to the quartic geometrically, as the intersection of two parabolas.

Abel proved in 1820 or thereabout that there is no general solution to the quintic equation (and by extension, no general solution to polynomial equations of degree 5 or higher). Sad but true.

Answer 3

Yes for the quartic. The general solution was one of the highlights of Medieval mathematics. See

http://mathworld.wolfram.com/QuarticEquation.html

Surprisingly, it's no for the quintic and higher degree polynomials. This was the work of Galois the night before his duel which killed him at age 20. See

http://mathworld.wolfram.com/QuinticEquation.html

which also details what can be done.

Answer 4

There is a formula for the quartic equation(though very complicated)
But there is not and can´t be a formula for quintic equation

Answer 5

A lot of people believe that there is "no general solutIon" for quintic equations, but there actually are. There is no general solution for quintic equation in terms of ordinary algebraic radicals, but there is in terms of "Bling radicals". Check the wiki on Bling radicals, as well as the wiki on the quintic equation, to see that a general solution is indeed possible.

Answer 6

you can find one/two roots
by trial and error method
then use long/synthetic division
to simplify the given equation
and factorise it further

Answer 7

i think that it is only thru synthetic division that you can solve your problem.