Is there an easier way to solve rational equations?

[cmcraes]

Hi there, I've been learning about Solving for x in rational expressions with Mono, Bi and tri nomial denominators, and I was curious if there was an easier way to solve them than by factoring.

I already understand and fully grasp the concept, but its just annoying to go through all the same steps over and over again.

Is there some process (maybe integration, or derivation) that could 'simplify' the problem? Thanks!

 

[HallsofIvy]

You multiply on both sides of the equations by the "least common denominator" to convert to a polynomial equation. Then you have to solve the polynomial equation- and one way of doing that is factoring! I presume you know the "quadratic formula". There also exist formulas (though much more difficult http://www.sosmath.com/algebra/factor/fac11/fac11.html, http://www.sosmath.com/algebra/factor/fac12/fac12.html). There do not, and can not, exist formula that solve higher degree polynomial equations (in terms of elementary functions such are roots).

 

[cmcraes]

Yes i know you can turn it into a Polynomial equation and factor it, Thats exactly what I am trying to find a quicker way of doing! Haah

 

[Mark44]

I don't think there is a quicker way. When there is a polynomial in the denominator, it's much more convenient to have the polynomial in factored form (as a product of lower-degree polynomials) than it is to have it as a sum of terms.

 

[DrewD]

I wouldn't be surprised if you could come up with a formula for a few certain classes of rational equations (like you can for quadratic, cubic, quartic), but it would not necessarily be any easier to solve. It might be fun to find some... but I'm not going to do it.

 

[maistral]

I would run multiple Newton iterations then deflate the polynomial.

Tough luck if what comes out of there is not an easy fraction.