The distributive law
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$ , but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
algebra-precalculus arithmetic
$endgroup$
add a comment |
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$ , but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
algebra-precalculus arithmetic
$endgroup$
$begingroup$
It is also the distributive law or rule.
$endgroup$
– Dr. Sonnhard Graubner
Dec 14 '18 at 16:19
$begingroup$
This is distribution done twice.
$endgroup$
– Randall
Dec 14 '18 at 16:20
add a comment |
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$ , but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
algebra-precalculus arithmetic
$endgroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$ , but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
algebra-precalculus arithmetic
algebra-precalculus arithmetic
asked Dec 14 '18 at 16:18
Ashraf BenmebarekAshraf Benmebarek
485
485
$begingroup$
It is also the distributive law or rule.
$endgroup$
– Dr. Sonnhard Graubner
Dec 14 '18 at 16:19
$begingroup$
This is distribution done twice.
$endgroup$
– Randall
Dec 14 '18 at 16:20
add a comment |
$begingroup$
It is also the distributive law or rule.
$endgroup$
– Dr. Sonnhard Graubner
Dec 14 '18 at 16:19
$begingroup$
This is distribution done twice.
$endgroup$
– Randall
Dec 14 '18 at 16:20
$begingroup$
It is also the distributive law or rule.
$endgroup$
– Dr. Sonnhard Graubner
Dec 14 '18 at 16:19
$begingroup$
It is also the distributive law or rule.
$endgroup$
– Dr. Sonnhard Graubner
Dec 14 '18 at 16:19
$begingroup$
This is distribution done twice.
$endgroup$
– Randall
Dec 14 '18 at 16:20
$begingroup$
This is distribution done twice.
$endgroup$
– Randall
Dec 14 '18 at 16:20
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Because:
$$left (x+y right) left (x+y right) = underbrace{(x+y)+(x+y)+...(x+y)}_{x + ytext{ times}}$$
$$=underbrace{(x+y)+(x+y)+...(x+y)}_{xtext{ times}}+underbrace{(x+y)+(x+y)+...(x+y)}_{ytext{ times}}$$
$$=underbrace{x}_{xtext{ times}}+underbrace{y}_{xtext{ times}}+underbrace{x}_{ytext{ times}}+underbrace{y}_{ytext{ times}}$$
$$=xx+xy+yx+yy$$
$endgroup$
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
add a comment |
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$
This works because $4$ is a (positive) integer; but for non-integer factors you can't use the "repeated addition"-argument.
but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
The distributive law works even if $x$ and/or $y$ are not integers.
Keep one of the factors together in a first step, and apply distributivity twice:
$$begin{align}
(color{blue}{x}+color{red}{y})(x+y) & =color{blue}{x}(x+y)+color{red}{y}(x+y) \ & =color{blue}{x}x+color{blue}{x}y+color{red}{y}x+color{red}{y}y
end{align}$$
$endgroup$
add a comment |
$begingroup$
Do it in steps.
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time: Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement. You should do it directly.
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$ and $(x + y)y = xy + yy$. So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
$endgroup$
add a comment |
Your Answer
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3 Answers
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active
oldest
votes
3 Answers
3
active
oldest
votes
active
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votes
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oldest
votes
$begingroup$
Because:
$$left (x+y right) left (x+y right) = underbrace{(x+y)+(x+y)+...(x+y)}_{x + ytext{ times}}$$
$$=underbrace{(x+y)+(x+y)+...(x+y)}_{xtext{ times}}+underbrace{(x+y)+(x+y)+...(x+y)}_{ytext{ times}}$$
$$=underbrace{x}_{xtext{ times}}+underbrace{y}_{xtext{ times}}+underbrace{x}_{ytext{ times}}+underbrace{y}_{ytext{ times}}$$
$$=xx+xy+yx+yy$$
$endgroup$
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
add a comment |
$begingroup$
Because:
$$left (x+y right) left (x+y right) = underbrace{(x+y)+(x+y)+...(x+y)}_{x + ytext{ times}}$$
$$=underbrace{(x+y)+(x+y)+...(x+y)}_{xtext{ times}}+underbrace{(x+y)+(x+y)+...(x+y)}_{ytext{ times}}$$
$$=underbrace{x}_{xtext{ times}}+underbrace{y}_{xtext{ times}}+underbrace{x}_{ytext{ times}}+underbrace{y}_{ytext{ times}}$$
$$=xx+xy+yx+yy$$
$endgroup$
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
add a comment |
$begingroup$
Because:
$$left (x+y right) left (x+y right) = underbrace{(x+y)+(x+y)+...(x+y)}_{x + ytext{ times}}$$
$$=underbrace{(x+y)+(x+y)+...(x+y)}_{xtext{ times}}+underbrace{(x+y)+(x+y)+...(x+y)}_{ytext{ times}}$$
$$=underbrace{x}_{xtext{ times}}+underbrace{y}_{xtext{ times}}+underbrace{x}_{ytext{ times}}+underbrace{y}_{ytext{ times}}$$
$$=xx+xy+yx+yy$$
$endgroup$
Because:
$$left (x+y right) left (x+y right) = underbrace{(x+y)+(x+y)+...(x+y)}_{x + ytext{ times}}$$
$$=underbrace{(x+y)+(x+y)+...(x+y)}_{xtext{ times}}+underbrace{(x+y)+(x+y)+...(x+y)}_{ytext{ times}}$$
$$=underbrace{x}_{xtext{ times}}+underbrace{y}_{xtext{ times}}+underbrace{x}_{ytext{ times}}+underbrace{y}_{ytext{ times}}$$
$$=xx+xy+yx+yy$$
edited Dec 14 '18 at 16:30
answered Dec 14 '18 at 16:24
Bram28Bram28
62.9k44793
62.9k44793
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
add a comment |
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
If $x+y$ isn't an integer, what does "$x+y$ times" mean then? The same for $x$ and $y$ separately.
$endgroup$
– StackTD
Dec 14 '18 at 16:30
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
@StackTD Excellent question! Obviously one needs to do something else in that case ... but I figured this would be in the spirit of the OP
$endgroup$
– Bram28
Dec 14 '18 at 16:32
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
$begingroup$
I'm not sure what OP means, but my comment is intended to make OP think about that too because even in his first example, $x$ and $y$ need not be integers (and the argument still holds, thanks to the 4!).
$endgroup$
– StackTD
Dec 14 '18 at 16:33
add a comment |
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$
This works because $4$ is a (positive) integer; but for non-integer factors you can't use the "repeated addition"-argument.
but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
The distributive law works even if $x$ and/or $y$ are not integers.
Keep one of the factors together in a first step, and apply distributivity twice:
$$begin{align}
(color{blue}{x}+color{red}{y})(x+y) & =color{blue}{x}(x+y)+color{red}{y}(x+y) \ & =color{blue}{x}x+color{blue}{x}y+color{red}{y}x+color{red}{y}y
end{align}$$
$endgroup$
add a comment |
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$
This works because $4$ is a (positive) integer; but for non-integer factors you can't use the "repeated addition"-argument.
but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
The distributive law works even if $x$ and/or $y$ are not integers.
Keep one of the factors together in a first step, and apply distributivity twice:
$$begin{align}
(color{blue}{x}+color{red}{y})(x+y) & =color{blue}{x}(x+y)+color{red}{y}(x+y) \ & =color{blue}{x}x+color{blue}{x}y+color{red}{y}x+color{red}{y}y
end{align}$$
$endgroup$
add a comment |
$begingroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$
This works because $4$ is a (positive) integer; but for non-integer factors you can't use the "repeated addition"-argument.
but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
The distributive law works even if $x$ and/or $y$ are not integers.
Keep one of the factors together in a first step, and apply distributivity twice:
$$begin{align}
(color{blue}{x}+color{red}{y})(x+y) & =color{blue}{x}(x+y)+color{red}{y}(x+y) \ & =color{blue}{x}x+color{blue}{x}y+color{red}{y}x+color{red}{y}y
end{align}$$
$endgroup$
$4left (x+y right)=4x+4y $ because $4left (x+y right) =left (x+y right) +left (x+y right) +left (x+y right) +left (x+y right)$
This works because $4$ is a (positive) integer; but for non-integer factors you can't use the "repeated addition"-argument.
but why is $left (x+y right) left (x+y right) =xx+xy+yx+yy$?
The distributive law works even if $x$ and/or $y$ are not integers.
Keep one of the factors together in a first step, and apply distributivity twice:
$$begin{align}
(color{blue}{x}+color{red}{y})(x+y) & =color{blue}{x}(x+y)+color{red}{y}(x+y) \ & =color{blue}{x}x+color{blue}{x}y+color{red}{y}x+color{red}{y}y
end{align}$$
answered Dec 14 '18 at 16:20
StackTDStackTD
22.9k2151
22.9k2151
add a comment |
add a comment |
$begingroup$
Do it in steps.
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time: Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement. You should do it directly.
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$ and $(x + y)y = xy + yy$. So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
$endgroup$
add a comment |
$begingroup$
Do it in steps.
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time: Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement. You should do it directly.
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$ and $(x + y)y = xy + yy$. So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
$endgroup$
add a comment |
$begingroup$
Do it in steps.
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time: Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement. You should do it directly.
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$ and $(x + y)y = xy + yy$. So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
$endgroup$
Do it in steps.
You accept that $M(x+y) = Mx + My$
So replace $M$ with $(x+y)$ and you get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y$
Now distribute a second time: Replace $x$ with $A$ and $y$ with $B$ to get:
$(x+y)(x+y) = M(x+y) =$
$Mx + My = $
$(x + y)x + (x+y)y=$
$(x+y)A + (x+y)B=$
$xA + yA + xB + yB =$
$xx + yx + xy + yy =$
$x^2 + 2xy + y^2$.
Of you you don't have to, and you shouldn't, do all that replacement. You should do it directly.
$(x+y)(x+y)=$ we treat one of the $(x+y)$ as a single thing and distribute across the other $x + y$.
$(x+y)(x+y) = (x+y)x + (x+y)y=$.
Now we have to sums to distribute: $(x+y)x = xx + yx$ and $(x + y)y = xy + yy$. So puting them together:
$(x+y)(x+y) = (x+y)x + (x+y)y=xx + yx + xy + yy=$.
And then some clean up:
$=x^2 + 2xy + y^2$
answered Dec 14 '18 at 16:51
fleabloodfleablood
71k22686
71k22686
add a comment |
add a comment |
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$begingroup$
It is also the distributive law or rule.
$endgroup$
– Dr. Sonnhard Graubner
Dec 14 '18 at 16:19
$begingroup$
This is distribution done twice.
$endgroup$
– Randall
Dec 14 '18 at 16:20