why does x^2 (x+2)-(x+2) factor into (x+2)(x^2-1)? *Better formatted in post.


Like, where does the “-1” come from? I feel like I am missing something painfully obvious…

Better format:

Factor x^(2)(x + 2) – (x + 2) = (x + 2)(x^(2) – 1) = (x + 2)(x – 1)(x + 1)

I understand that you’re supposed to factor out the (x+2), but for some reason I can’t grasp the sequence of orders happening there…

In: 0

On the left term, the x^2 is obvious factor. On the right term, 1 is a factor, because (x+2)*1 = (x+2). You could write it (x^2 ) (x+2) + (-1) (x+2).

When you factor out the (x+2) you’re having to divide the remaining components by that value.

x^(2)(x+2) / (x+2) is just x^2

-(x+2) / (x+2) is just -1

Ergo, (x+2)(x^(2)-1).

Have you learned factoring by grouping because that can be used here as well. If not, how you do it is you group a polynomial into parts that have a common factor.

So in this case, you would expand it into x^3 + 2x^2 – x – 2. Then group it into (x^3 – x) + (2x^2 – 2). You then factor out x from the first group leaving you with x (x^2 – 1). Then the second group factor out 2 leaving 2(x^2 – 1).

Once you have x(x^2 – 1) + 2(x^2 – 1) you factor out the (x^2 – 1) and group together the x +2 to get a final answer of (x^2 – 1)(x + 2).

A shortcut that doesn’t require expanding would be recognizing that the function is x^2 (x + 2) – 1 (x + 2) which would allow you to factor out (x + 2) right away and skip grouping but grouping is a good skill to have.