Factoring 4th degree polynomials.

[cp255]

So I needed to factor -4x⁵-8x⁴+8x³+4x.
I factored out a -4x and I am left with x⁴+2x³-2x²-4.

The problem is I am unsure how to factor x⁴+2x³-2x²-4.
I know how to long divide polynomials but I have not done synthetic division in over 4 years. From what I have seen on the internet it seems like a lot of guess and check.

 

[jbunniii]

You made a mistake when you divided by -4x. After dividing, the last term should be -1, not -4.

You can find a root by trial and error, or by using the rational root theorem: http://en.wikipedia.org/wiki/Rational_root_theorem

 

[epenguin]

Easier, it makes sense to group the pair with factor 2 together - you can certainly factor that pair. The remaining two terms have a factorisation that is very like one you should remember - since you say you are out of practice perhaps the only one you'd remember.

 

[eddybob123]

There is a quartic formula which you can use, and once you know its solutions, you know its factors. The only porblem is that it is extremely not recommended...
http://www.curtisbright.com/quartic/quartic-png.html

 

[CellCoree]

I understand your frustration with factoring 4th degree polynomials. It can be a challenging and time-consuming process, especially without using synthetic division. However, there are some techniques and strategies that can make factoring polynomials easier.

One approach is to look for common factors. In this case, we can see that all the terms have a factor of 2, so we can factor it out to get 2(x4+2x3-2x2-4).

Another technique is to try grouping the terms. We can group the first two terms and the last two terms to get x3(x+2)-2(x+2). Then we can factor out (x+2) to get (x3-2)(x+2).

If these methods do not work, we can also use the rational root theorem to find possible rational roots of the polynomial. By testing these possible roots, we can often find one that works and then use synthetic division to factor the polynomial.

Overall, factoring 4th degree polynomials can be a challenging task, but with practice and the use of different strategies, it can become easier. It is important to remember that there is not always a simple or quick solution, and sometimes a bit of trial and error is necessary. Keep exploring and trying different methods, and you will eventually be able to factor polynomials efficiently.