#### Keeyn

##### New member
Hi guys, I have a simple question but it doesn’t seem that simple. I understand that and why 3(x+4) times 2(x+3) gives me (3x+12) times (2x+6) with the brackets written. Indeed, brackets are important here. However, what I don’t understand is why 3(x+4)+2(x+3) (note that only the multiplication sign is replaced by the addition sign) gives 3x+12+2x+6 straight without the brackets ( I know that we do 3(x+4) first then 2(x+3) ). Isn’t it supposed to be expanded to (3x+12) + (2x+6) with the brackets just like in the case of multiplication above? I thought since 3(x+4) and 2(x+3) are together in one expression and they do not stand alone, the brackets therefore are mandatory in the expanded form since 3(x+4) is viewed as a set and 2(x+3) another set?

#### Dr.Peterson

##### Elite Member
Hi guys, I have a simple question but it doesn’t seem that simple. I understand that and why 3(x+4) times 2(x+3) gives me (3x+12) times (2x+6) with the brackets written. Indeed, brackets are important here. However, what I don’t understand is why 3(x+4)+2(x+3) (note that only the multiplication sign is replaced by the addition sign) gives 3x+12+2x+6 straight without the brackets ( I know that we do 3(x+4) first then 2(x+3) ). Isn’t it supposed to be expanded to (3x+12) + (2x+6) with the brackets just like in the case of multiplication above? I thought since 3(x+4) and 2(x+3) are together in one expression and they do not stand alone, the brackets therefore are mandatory in the expanded form since 3(x+4) is viewed as a set and 2(x+3) another set?
Once you have expanded 3(x+4) + 2(x+3) to (3x+12) + (2x+6), the parentheses are no longer required; the result is equivalent to 3x+12+2x+6. That is, for any value of x, the two expressions have the same value.

You appear to be thinking that the parts of the expression retain some invisible link to their origin; they don't. As long as two expressions always have the same value, they are equivalent, no matter what you have done to get from one to the other.

Or perhaps you are just wondering about the difference between (3x+12)(2x+6) and(3x+12) + (2x+6). The reason you can drop parentheses in the latter is that all the operations connecting the terms are additions, so the associative property of addition applies. That is not true in the former, where we have a mix of additions and multiplication, and the distributive property applies.

Note that if we couldn't write 3x+12+2x+6, we couldn't then rearrange (using the commutative property) and combine like terms (using the distributive property). This step of dropping parentheses is essential to what we are typically doing in algebra.

#### Keeyn

##### New member
I do understand that the parenthesis can be dropped and ultimately it leads to the same end results. What I don’t understand is that when teachers teach, they often miss out this step: Like in the case of 3(x+4)-6(3+x)...They miss out (3x+12)+(-18-6x), they jump straight to 3x+12-18-6x. Is that middle step non-existent? Is that something made up by my mind?

#### Dr.Peterson

##### Elite Member
I do understand that the parenthesis can be dropped and ultimately it leads to the same end results. What I don’t understand is that when teachers teach, they often miss out this step: Like in the case of 3(x+4)-6(3+x)...They miss out (3x+12)+(-18-6x), they jump straight to 3x+12-18-6x. Is that middle step non-existent? Is that something made up by my mind?
I don't think I can tell you why (or even whether) the majority of teachers would skip that particular step. That's up to them. Most likely it would be because they are talking to students whom they expect to have passed that hurdle, so that they focus their attention on other issues. I suppose I probably would skip it for most students, though of course not the first time I did an example like this.

I hope I made it clear in my response to you that the step with parentheses is entirely valid. It is just such a small step to the next form that we often don't bother to write it.

Actually, in your new example (which is considerably trickier than the first), there are many other steps that might be shown, if one wanted to show every detail:

3(x + 4) - 6(3 + x)
3(x + 4) + -[6(3 + x]
3(x + 4) + -1[6(3 + x)]
3(x + 4) + (-1*6)(3 + x)
3(x + 4) + (-6)(3 + x)
[3*x + 3*4] + [(-6)3 + (-6)x]
[3x + 12] + [-18 + -6x]
3x + 12 + -18 + -6x
3x + -6x + 12 + -18
(3 + -6)x + (12 + -18)
-3x + -6
-3x - 6

I would never write out all those steps; but that is what I am actually doing in simplifying the expression. We must omit details at some point in order to communicate clearly; and that requires assuming something about the reader or student.

The trouble is that any teacher of a group of students (and especially any author writing to many students they will never even see) can't know what each of them does or does not already know, and therefore will make some choices that are not suitable for all of them.

That's part of the reason I prefer tutoring individually. But even then, I have to find out what you are ready for by trying things out and seeing what results I get.

But back to your question, no, it is not your imagination. The step exists; it is just one that teachers may easily assume students can figure out for themselves.