Hey vermillion.
There is a result that allows you to factor general fourth degree polynomial (which you can assume to be true for algebraic problems):
Quartic function - Wikipedia, the free encyclopedia
hey guys this is probably pretty basic stuff for most of you. im however struggling with this.
i don't have any standard procedure for this and i can't figure out how its done. i just don't know what kind of method i should be searching for.
in my course it was said you are already supposed to know how to do this so im lacking skill on this. here is the problem:
i underlined the equation where the polynomial is factorized. how to do this? whats the method called and how can i learn to do this on any generic 4th degree polynomial?
Hey vermillion.
There is a result that allows you to factor general fourth degree polynomial (which you can assume to be true for algebraic problems):
Quartic function - Wikipedia, the free encyclopedia
thanks for your response. but i am afraid this article doesn't really help me here. in the article there is a different last term, an e. I bet there is an easier method out there as well. i hope someone can give another method.
and also in this case the a is always 1. i bet there is some kind of fancy polynomial trick for this.
The is factorized by finding the roots of
If it has any integer root it must be ±1 or ±2 or ±4 (actually we can skip the positive solutions since we will have all positive members on the equation so they can't give zero)
Here we see that -2 works and is a root.
Then dividing with x+2 we get which has roots -1-i and -1+i.
Ergo its factorization is (x+2)(x+1+i)(x+1-i)
You can use the same method with the other equation also.
It's not a general method. I.e it will not work in most cases. For that you have to use the link chiro gave.
Well there is a "fancy" method for a=1:
The factorization is where can be found from here: quartic formula | planetmath.org
very fancy indeed ok thanks guys. However i found out i was overcomplicating as apparently in the exams there won't be any cases where you have to solve 4th degree polynomials, in the worst case they are 3th and i can do that. Also i found a note in the lectures that states 4th degrees should be solved numerically but we won't be using calculators during exams so im good.
Also you might want to consider the guessing method were you guess a root and then use long division to go from say cubic to quadratic (in which case you use the quadratic formula or further guesses).
So for example if I had say (x-1)(x-2)^2
= (x-1)(x^2 - 4x + 4)
= (x^3 - 4x^2 + 4x - x^2 + 4x - 4)
= x^3 - 5x^2 + 8x - 4
If you tried a test solution of x = 1 you would get zero and thus you could factor (x-1) straight out.