I have a question I hope someone can explain it to me.

Question

We are working on Demonstrate that f actoring a polynomial. I was ask Why can you factor x² - 4 but you cannot factor x² + 4? How can you tell quickly which ones you can factor and which you cannot? I am not sure what to answer here. I feel like it has some thing to do with the GCF but I am sure can some just give me an example and explain to me.

Answer 1

actually, one can factor x^2 + 4

(x+2i)(x-2i) are the factors, where i is the sqrt of -1.

The reality is that you cant look at a polynomial and know how many real factors there are. Experience helps, but when one has

x^3+3x^2-3x-14 there is no easy way to know without some detailed examination.

Answer 2

The X^2 - 4 can be factored because the negative sign allows us to make one factor + and the other one - as in
(x+2)(x-2) which when we expand it has the middle term cancel--to wit:x^2 +2x-2x-4.
When we try to factor x^2+4, we KNOW both factors must be EITHER + or - because + x + = + and - x - = +. As long as the two factors have the same sign, we always get a middle term BUT x^2+4 has no middle term.

Answer 3

So if I understanding this for example I have x^2-6 I can factor because the negative sign I can get a middle term
which would be
(x+2)(x-3) expand it to
(x^2 +2x-3x-6)
(x^(2)-x-6)

With x^2+6 I could not get a middle term because the sign is positive which shows me that because When you multiply two + togther you a + and when you mulitiply two - togther you get a + so this can not have a middle term

Did I understand this correct