Posted by TOM on Tuesday, November 11, 2008 at 7:43pm.
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.
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.
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