Factori'z'?tion

[Yuseph]

Yo wussup guys

As you know, if I dont cover all the details i dont feel like i master the topic.

In this picture the author chose -1 as the hcf. But it confuses me because it so doesn't look natural and on top of that this is in contradiction with all the exercices he showed before.
To me 1 would be the natural hcf. Not -1.
I know he did it on purpose because he wanted both x + 3 to match but isnt there a rule that make it obligatory to migrate elements into the brackets - here x + 3 - without changing their positive or negative sign ?

 

[Dr.Peterson]

I think you're saying you understand what he did, and why; but I'm not sure what you think is wrong. What is "obligatory"? Generally, we are not obligated to do anything; we choose to do what is (a) valid, and (b) useful for our goals.

The term "HCF" (or "GCF" for me) has slightly different meanings in different contexts. In talking about numbers on their own, you would only consider positive numbers; but here, in a polynomial and with negative coefficients, it is appropriate to call it -1.

Generally, when I factor the "GCF" from a polynomial, I want the remaining polynomial to have a positive leading term, because that is useful in many situations. So it is a good habit, when the leading term is negative, to take the sign as part of the "GCF" that you factor out. In this case, it makes the two (x+3)'s match, just as you said.