How to Factor Polynomial Equations

[The Matador]

I don't know why but I have a lot of trouble factoring. After 3 years of it I would expect to be a lot better at it but I still find it as hard as when I first learned it. I am just wondering if anyone here can help me understand it a little different then my teachers have.

To date I think I finally understand stuff like
$$2x^2-2x-12$$
$$x^3-5x^2-6x$$
$$5x^3-45x$$

The stuff I am having trouble with is things like this
$$x^3+x^2-4x-4$$
$$x^8-1$$
$$2m(m-n) + (m+n)(m-n)$$

Is there any certain rules or guidelines I may be able to follow to make these kind of questions easier. Right now I am not really sure how to start them.

 

[cristo]

For the first one if you can't factor it, then guess a factor and divide the polynomial by that factor to get a quadratic.

The second one seems pretty straightfoward... what have you tried?

In the third one, what is the common factor in both terms?

 

[The Matador]

The first one I understand now, I get (x+1)(x+2)(x-2)

The second one I am still not really sure

Here is the third one I think
(m-n)[2m + (m+n)]
(3m+n)(m-n)

 

[rock4christ]

I don't know why but I have a lot of trouble factoring. After 3 years of it I would expect to be a lot better at it but I still find it as hard as when I first learned it. I am just wondering if anyone here can help me understand it a little different then my teachers have.

To date I think I finally understand stuff like
$$2x^2-2x-12$$
$$x^3-5x^2-6x$$
$$5x^3-45x$$

The stuff I am having trouble with is things like this
$$x^3+x^2-4x-4$$
$$x^8-1$$
$$2m(m-n) + (m+n)(m-n)$$

Is there any certain rules or guidelines I may be able to follow to make these kind of questions easier. Right now I am not really sure how to start them.

I HATE factoring. I am great at all math I have ever tried(literally) and got a D on a factoring test. any thing beyond the complexity of x² +ax +b i fail at. once x² has a coefficient I fail miserably

 

[turdferguson]

In order to understand the 3rd line better, Ill explain another way to factor the 1st line. You can group x^3 with x^2 and -4x with -4. Then you can factor each piece and youre left with:
x^2(x+1) -4(x+1)
This form is very similar to that third line. Next, you treat x^2 and -4 as coeffieients and factor out an (x+1) from both terms:
(x+1)(x^2 - 4) This can then be reduced to what you got, and the third line also looks good

But youre having the most trouble with the second line, x^8 - 1. Both terms are perfect squares. Whats the square root of x^8? Are you done after that, or can it be factored further?

 

[AlephZero]

For problems where you know there is an easy answer (otherwise you wouldn't have been asked the question!) this often helps:

If you want to factor $ax^n \pm \dots \pm b$ then for a factor of the form $px \pm q$ it is a good bet that p is a factor of a, and q is a factor of b.

Also, use the remainder theorem: if (x-a) is a factor, the polynomial is zero when x = a. So in the $x^3+x^2-4x-4$ example it's fairly obvious the polynomial is zero when x = -1 therefore (x+1) is a factor.