please take me step by step.
Factor: $$x^{4}-12x^{3}+59x^{2}-138x+130$$

Question

please take me step by step.
Factor: $$x^{4}-12x^{3}+59x^{2}-138x+130$$

*insert name here* · Answer

What attempts have you made?

mokeira · Answer

i was told i should do something like
$$(x^{2}-ax+b)(x^{2}+cx+d)$$ but i have no idea which should be a in such a case

campbell_st · Answer

I think is 

(x^2 - 6x + 13)(x^2 - 6x + 10)

just trial and error

anonymous · Answer

So, let me tell you how to start of this pain in the butt of a problem ( I don't really wanna do it all :P) 

Your hint is absolutely correct, I can show you how to eliminate possibilities using logic.

The easiest coefficients to start with here are c and d, because they are affected by the least amount of factors when expanding, thus will be the easiest to think about when factoring. Because in the polynomial, 130 is a constant. This means it is created when multiplying two factor polynomials which contain constants that are also a multiples of factors of 130. It is harder to explain in words but I am saying that 
c*d=130, therefore c and d are a made up of {2,5,13}. This means that c and d are limited to theses sets of numbers: {10,13} or {2,65} or {5,26}. 

The next easiest one to look at is the x cubed term. We know that its coefficient is only manipulated by the two factors exactly as such: c-a=12

The next step is to look at the term with x in it. We know this term's coefficient can ONLY be manipulated by -a,b,c,d exactly as such: (-a*d)+b*c=-138. 

The final and most difficult term to analyze is the x squared term. We know that its coefficient can be manipulated by a variety of terms. I will spell out every term to make this one very clear. When you expand those to factors you have: 
dx^2 +bx^2 -acx^2=59x^2. 
So the coefficient is manipulated exactly as such: d+b-ac=59

So what did I just do for you? I created a big system of equations that you can solve for each of these coefficients. Lets go over the four equations
$$c*d=130$$
$$c-a=12$$
$$ (-a*d)+b*c=-138$$
$$d+b-ac=59$$

Now go forth and solve!! :D

anonymous · Answer

btw just ignore the sets thing (i kinda switched streams halfway because of the how the coefficients were intertwined.)