Help w/ understanding concept of distributing a neg. sign

[Teabreeze]

Homework Statement

Basically, I have to distribute this negative sign in this math expression:

- (-x^2 + 12x +24)

Homework Equations

none

The Attempt at a Solution

I know from the rules that we get:

(x^2 - 12x - 24)

My question is why do we distribute the negative sign to begin with? I thought we treat everything in parenthesis in math as a whole unit. So why do we go in and distribute the negative sign to EACH term?

tym

 

[Mark44]

Homework Statement

Basically, I have to distribute this negative sign in this math expression:

- (-x^2 + 12x +24)

Homework Equations

none

The Attempt at a Solution

I know from the rules that we get:

(x^2 - 12x - 24)

My question is why do we distribute the negative sign to begin with? I thought we treat everything in parenthesis in math as a whole unit. So why do we go in and distribute the negative sign to EACH term?

tym

The simplest way to look at this is by considering that the minus sign out front is -1. So $-(-x^2 + 12x + 24) = -1(-x^2 + 12x + 24)$. Using the distributive law, the last expression becomes $x^2 - 12x - 24$.

 

[Buzz Bloom]

Hi Teabreeze:

There is not any requirement that distributing the minus sigh is necessary, but sometimes it is convenient, and it is useful to have an easy to understand rule when one wants to take advantage of the convenience. An example of when it might be convenient is when you have a more complicated expression you want to simplify, e.g.,

- (- 3x² - 12x - 24) + (4x² - (2x - 6))​

In this case distributing minus signs can help you avoid mental errors while doing the simplification.

Regards,
Buzz

 

[member 587159]

You know distributivity?
a(b + c) = ab + bc

Now if a = -1, then -1(b + c) = -b - c
We just do not write the '1'. We write -(a+b) instead of -1(a + b).

Also, think about it with examples.
For example, -2(3 + 4) = -6 - 8 = -14

 

[Teabreeze]

You know distributivity?
a(b + c) = ab + bc

Now if a = -1, then -1(b + c) = -b - c
We just do not write the '1'. We write -(a+b) instead of -1(a + b).

Also, think about it with examples.
For example, -2(3 + 4) = -6 - 8 = -14

You wrote:

a(b+c) = ab + bc

BUT, I thought distributive property was this instead:
a(b+c) = ab +ac

Is this the correct version or is yours the correct one?

Thanks.

 

[DrClaude]

You wrote:

a(b+c) = ab + bc

BUT, I thought distributive property was this instead:
a(b+c) = ab +ac

Is this the correct version or is yours the correct one?

Yes. @Math_QED made a mistake (typo I guess). The numerical example is correct.

 

[Teabreeze]

Hi Teabreeze:

There is not any requirement that distributing the minus sigh is necessary, but sometimes it is convenient, and it is useful to have an easy to understand rule when one wants to take advantage of the convenience. An example of when it might be convenient is when you have a more complicated expression you want to simplify, e.g.,

- (- 3x² - 12x - 24) + (4x² - (2x - 6))​

In this case distributing minus signs can help you avoid mental errors while doing the simplification.

Regards,
Buzz

I think I'm confused, because I thought we literally couldn't distribute. In other words, I thought at that having some expression in parenthesis meant that that WHOLE thing was a "unit" and whatever you do to it, then you must do it to the whole thing.

So, when you distribute the way I did in the original example, it's "going into the unit" and placing a minus sign in front of each term. But that to me means we're messing with the parenthesis. I thought that parenthesis means you HAVE to treat something as a whole unit. So if you want to subtract that unit (everything inside the parenthesis), then you have to subtract the value of the WHOLE thing.

How can you put a minus sign onto each individual part of the whole thing and have that work? So still a bit confused guys. Am I missing something?

 

[SammyS]

I think I'm confused, because I thought we literally couldn't distribute. In other words, I thought at that having some expression in parenthesis meant that that WHOLE thing was a "unit" and whatever you do to it, then you must do it to the whole thing.

So, when you distribute the way I did in the original example, it's "going into the unit" and placing a minus sign in front of each term. But that to me means we're messing with the parenthesis. I thought that parenthesis means you HAVE to treat something as a whole unit. So if you want to subtract that unit (everything inside the parenthesis), then you have to subtract the value of the WHOLE thing.

How can you put a minus sign onto each individual part of the whole thing and have that work? So still a bit confused guys. Am I missing something?

Do you have a problem with distributing the 2 in $\ 2(x+3) \$ which then gives you $\ 2x+6 \ ?$

 

[SteamKing]

I think I'm confused, because I thought we literally couldn't distribute. In other words, I thought at that having some expression in parenthesis meant that that WHOLE thing was a "unit" and whatever you do to it, then you must do it to the whole thing.

So, when you distribute the way I did in the original example, it's "going into the unit" and placing a minus sign in front of each term. But that to me means we're messing with the parenthesis. I thought that parenthesis means you HAVE to treat something as a whole unit. So if you want to subtract that unit (everything inside the parenthesis), then you have to subtract the value of the WHOLE thing.

How can you put a minus sign onto each individual part of the whole thing and have that work? So still a bit confused guys. Am I missing something?

Yes. The parentheses are there just to clarify the expression - to make sure that readers understand what goes with what.

For example, if some one were to write 3 - 2/4 +6, then it is unclear what is meant. One could say that 3 - 2/4 + 6 = 9.5, but then again
if one sees (3 - 2) / (4 + 6), then it becomes clear that this is equivalent to 1/10 and nothing else.

Still, one is able to take the negative of such an expression as - (3 - 2) / (4 + 6) = (-3 - (-2)) / ( 4 + 6) = (-3 + 2) / (4 + 6) = -1/10.

This "unit" interpretation which you seem to think applies breaks down if you have an expression like (a + b) ⋅ (c + d).

The rules of algebra give the product (a + b) ⋅ (c + d) = ac + ad + bc + bd, which otherwise could not be obtained if you were not permitted to distribute multiplication over addition.