Factoring a higher order polynomial

[mateomy]

Homework Statement

$$x^4 + 4x^3 - x^2 + 16x - 12$$

I know that with some higher order polynomials you can substitute say x^4 as a = x^2 thereby making it easier to break the thing apart and find its factors. I know I am looking for 4 roots, but my little substitution method doesn't really work here. Is there another way that this can be done? Another formula that I am overlooking?

 

[eumyang]

Well, there's the Rational Zeros Theorem, but it's not going to help if none of the real zeros are rational. (Assuming the original problem was typed correctly, if you graph this on my calculator, you'll see what I mean.) Besides that, within the realm of Elementary Algebra (as opposed to Linear Algebra or Abstract Algebra), I'm not aware of any other method to use.

 

[tiny-tim]

hi mateomy!

are you sure it isn't minus 4x³?

 

[mateomy]

Dang, you're right, I am looking at the problem again and it is a
$$-4x^3$$

do you know something I (clearly) dont?

Haha.

And this is a "brain teaser" handout my teacher gave us which only deals with math up to precalc. Its not for credit or anything, I am just lookin at some of these problems with my WTF button pushed down...

Thanks for the pointers everyone.

 

[symbolipoint]

The Rational Roots theorem mentioned from eumyang is probably the right way to try. You want to try division by different binomials based on what this theorem suggests and you should find some results of divisions which give no remainder.

 

[mateomy]

Alright, sounds good. Thanks again for the pointers everyone.