Distributive property

I

ichbi

New member
Joined
Nov 6, 2019
Messages
4
Hey guys,

After seven years(!) I have to take up mathematics again for my masters degree. In secondary school I wasn't great at it, so I just started practicing again. I'm using a book which isn't providing a lot of help. I hope you can help me to understand it.

The exercise which I'm doing now starts easy with (a+b) (c+d). I understand that the answer is ac + ad + bc + bd. I can solve it by drawing a visualization. But I don't understand the following: (a+b) (c-d). One plus changed into a minus. So, how do I now where to put the pluses and the minuses? Another one which is (in my opinion a bit harder): -(a-b) (c+d). Long story short: how do I know where to put the pluses and minuses between the ac, ad, bc bd?

There is also another exercise which is hard, but I don't even know how it is called in math.. a few of the questions are:
10xy + 5x * 3y - 12x =
18ab : 6a * b - 3 =
9a * 3c : 9c * 3a =

I don't need the answers, but I do need a help with how to solve these.
 
Hi, have you tried proceeding like this:
(a+b)*(c-d)=a(c-d)+b(c-d)=ac-ad+bc-bd
and -(a-b)*(c+d)=-[a(c+d)-b(c+d)]=-[ac+ad-bc-bd]=bc+bd-ac-ad
 
One of the odd realities of such an exercise to that negative sign has two different functions. It does represent subtractions, but it also serves as a sign of a number.

+only
(a+b)(c+d) = a(c+d) + b(c+d) = ac + ad + bc + bd
Nothing suspicious about that.

(a+b)(c-d) = a(c-d) + b(c-d) = ac - ad + bc - bd
If we do it one piece at a time, it is less confusing.

Some like to change things around a little and separate the two meanings. (c-d) = (c + (-d)) -- I've never been a fan of that. There are already too many signs floating around - too many errors to make. On the other hand, some find this beneficial.

(a-b)(c+d) = a(c+d) - b(c+d) = ac + ad - bc - bd
Just one piece at a time. No need to try to jump to the end. Let the notation help you.

-(a-b)(c+d) = -[a(c+d) - b(c+d)] = -[ac + ad - bc - bd] = -ac - ad + b c + bd
In this case, we saved that one "-" for the end, keeping it out of the way until we were ready for it.

Slowly. Systematically. Deliberately.
 
Actually, there are three uses for the minus sign. It is the most confusing symbol in beginning algebra.

It indicates the operation of subtraction. 8 - 3 indicates the operation of subtraction.

It indicates that a specific numeric value is less than zero. - 3 < 0.

It indicates that an unknown number is the additive inverse of a different unknown number, but does not mean that the unknown number is less than zero. - x means that x + (- x) = 0.

My advice is the opposite of tkhunny's and so is probably wrong, but I have found it has helped those I have tutored..

For a beginner, I do advise changing a - b, meaning a - (+b), to a + (-b) because they give an identical result.

So now your distributive rule works just as you memorized it.

[MATH](a - b)(c - d) = \{a + (-b)\}\{c + (-d)\} = a\{c + (- d)\} + (-b)\{c + (-(d)\}.[/MATH]
Now all you need to remember is this multiplication rule: (1) anything times zero is zero, (2) but, with no zeroes, (a) like times like is positive, and (b) like times unlike is negative.

So lets apply that rule.

[MATH](a - b)(c - d) = \{a + (-b)\}\{c + (-d)\} = a\{c + (- d)\} + (-b)\{c + (-(d)\} = \\ (a)(c) + (a)(-d) + (-b)(c) + (-b)(-d) = ac - ad - bc + bd.[/MATH]
 
JeffM said:
Actually, there are three uses for the minus sign. It is the most confusing symbol in beginning algebra.

It indicates the operation of subtraction. 8 - 3 indicates the operation of subtraction.

It indicates that a specific numeric value is less than zero. - 3 < 0.

It indicates that an unknown number is the additive inverse of a different unknown number, but does not mean that the unknown number is less than zero. - x means that x + (- x) = 0.

My advice is the opposite of tkhunny's and so is probably wrong, but I have found it has helped those I have tutored..

For a beginner, I do advise changing a - b, meaning a - (+b), to a + (-b) because they give an identical result.

So now your distributive rule works just as you memorized it.

[MATH](a - b)(c - d) = \{a + (-b)\}\{c + (-d)\} = a\{c + (- d)\} + (-b)\{c + (-(d)\}.[/MATH]
Now all you need to remember is this multiplication rule: (1) anything times zero is zero, (2) but, with no zeroes, (a) like times like is positive, and (b) like times unlike is negative.

So lets apply that rule.

[MATH](a - b)(c - d) = \{a + (-b)\}\{c + (-d)\} = a\{c + (- d)\} + (-b)\{c + (-(d)\} = \\ (a)(c) + (a)(-d) + (-b)(c) + (-b)(-d) = ac - ad - bc + bd.[/MATH]
Click to expand...
Actually similar to your method is how I was taught when I was a kid.
 
yoscar04 said:
Actually similar to your method is how I was taught when I was a kid.
Click to expand...
I recall such lessons, too. They just didn't speak to me, for the reason stated above, so I did my own thing. I know; I'm a rebel.
 
tkhunny said:
I recall such lessons, too. They just didn't speak to me, for the reason stated above, so I did my own thing. I know; I'm a rebel.
Click to expand...
Someone (I forget whom) said that correct answers do not care how you find them.
 
JeffM said:
… there are three uses for the minus sign. It is the most confusing symbol in beginning algebra …
Click to expand...
As a tutor, I've definitely seen a lot of confusion there. What I'd like to see is better distinction in beginning algebra between negation and subtraction. My TI-89 calculator has different buttons and uses different screen symbols for each. Unfortunately, most systems of writing use the same symbol for both: a hyphen.

8 - 3

That hyphen represents subtraction (i.e., it's an arithmetic operator).

-x

That hyphen represents negation (i.e., the negative sign represents a factor of -1).

Perhaps, stressing the difference in class would reduce confusion surrounding things like "adding the opposite" as a substitute for subtraction (because those hyphens mean two different things).

8 - 3 = 8 + (-3)

?
 
Otis said:
As a tutor, I've definitely seen a lot of confusion there. What I'd like to see is better distinction in beginning algebra between negation and subtraction. My TI-89 calculator has different buttons and uses different screen symbols for each. Unfortunately, most systems of writing use the same symbol for both: a hyphen.

8 - 3

That hyphen represents subtraction (i.e., it's an arithmetic operator).

-x

That hyphen represents negation (i.e., the negative sign represents a factor of -1).

Perhaps, stressing the difference in class would reduce confusion surrounding things like "adding the opposite" as a substitute for subtraction (because those hyphens mean two different things).

8 - 3 = 8 + (-3)

?
Click to expand...
It is one of the places where the historical development of notation makes sense but leaves the beginning student in an utter fog. The calculator I use has a negation key. Time the curriculum caught up.
 
I've seen some systems of instruction that introduce negation using a distinct symbol, [MATH]^-3[/MATH], and then much later point out that it can be written instead as [MATH]-3[/MATH], when students have enough experience to see that they can still understand it unambiguously.

In the long run, the multiple uses of the hyphen work well together; it's only initially that it's confusing.
 
JeffM said:
… Time the curriculum caught up.
Click to expand...
Yup. Another way instructors could initially reinforce negation would be to say "negative" instead of "minus".

-4 + (-x - 1) = -1

Good: "negative four plus the difference negative x minus 1 equals negative one"

Not as good: "minus four plus minus x minus 1 equals minus 1"

Okay … back to reality …

\(\;\)
 
Otis said:
Yup. Another way instructors could initially reinforce negation would be to say "negative" instead of "minus".

-4 + (-x - 1) = -1

Good: "negative four plus the difference negative x minus 1 equals negative one"

Not as good: "minus four plus minus x minus 1 equals minus 1"

Okay … back to reality …

\(\;\)
Click to expand...
You have sold me.
 
One of few things I recall from a textbook of MANY years ago... Trying to emphasize one meaning of the symbol "-".

1594920040127.png
With apologies to Dolciana, Berman, and Freilich (since I can't remember anyone else writing textbooks back then).
 
You must log in or register to reply here.
Top