Download the distributive law

ALGEBRA
Sec. 4
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THE DISTRIBUTIVE LAW
DEFINITION: one term
By ’a term’ we mean any one number, or variable, or a product and/or quotient of several numbers and/or variables.
The key is the absence of any addition or subtraction in a term. For example, 3xy is one term while 3x + y is NOT
a term since it is not the product or quotient of several numbers, in fact it is the sum of two terms.
IDEA
Thus far, our axioms have come in pairs. We have met the Times Table as well as the Addition Table, Associativity
for Addition as well as Associativity for Multiplication, similarly two commutativity axiom one for addition and
one for multiplication. This will be a reoccurring theme, yet the Distributive Law is an exception. We will NOT
learn two different distributive laws. The distributive law is, thus far, the only axiom which combines a statement
about both binary operations, addition and multiplication.
AXIOM: DISTRIBUTIVE LAW
< DL >
Having met the definition of term, we can begin to state our axioms in terms of terms. That is, unless stated
otherwise, these axioms will be adopted not just for the natural numbers, N, but for all other families of numbers
we will soon meet. Even more generally, they will be adopted for any type of term. A basic illustration of the
distributive law is the following: 3(2 + 5) = 3 · 2 + 3 · 5. A more general illustration would be one were we see it
applied not just to natural numbers but any sort of term such as
stuf f (blah + potato) = stuf f · blah + stuf f · potato
where stuf f, blah, and potato could be any term or sum of terms.
Another example:
twice
+
!
= twice
+ twice
and another,
lost
+
+ keys
!
= lost
+ lost
+ lost keys
Moreover, we will aslo accept the distributive law where the sum occurs on the left side such as (2 + 5)3 = 2 ·3 + 5 ·3.
Moreover, we will also accept it when more than two terms are being added, such as 2(3 + 1 + 5) = 2 · 3 + 2 · 1 + 2 · 5.
Formally, we state it, if A, B, and C represent each a term or sum of terms:
A(B + C) = AB + AC
< DL >
(B + C)A = BA + CA
< DL >
and
It should be noted that the distributive law does not apply to everything in the world. For example, you will soon
learn about functions, such as logs, exponential, or sine and cosine, these functions look like variables but are not.
For example And generally, the distributive law does not apply to functions. It is worth emphasizing that we will
assume the DL axiom for natural numbers, variables or symbols that represent natural numbers, as well as other
families of numbers which we will meet soon.
For now, here are some typical uses of the Distributive Law.
DL to get rid of Parenthesis We can use the distributive law to get rid of parenthesis.
3(x + 1) = 3 · x + 3 · 1
< DL >
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2007
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ALGEBRA
Sec. 4
MathHands.com
Márquez
DL to add Parenthesis We can use the distributive law to get rid of parenthesis.
3 · x + 3 · 1 = 3(x + 1)
< DL >
more generally,
blah · x + blah · 1 = blah(x + 1)
< DL >
or
blah · + blah · △ = blah( + △)
< DL >
DL to combine like-terms
3x2 + 5x2
=
=
(3 + 5)x2
2
8x
< DL >
< AT >
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2007
MathHands.com
ALGEBRA
Sec. 4
MathHands.com
Márquez
Some questions to think about
1. How many terms does this expression have?
3x + 1
A. 1
B. 2
C. 3
D. 4
E. none of the above
2. How many terms does this expression have?
3x + x
A. 1
B. 2
C. 3
D. 4
E. none of the above
3. How many terms does this expression have?
3x · 1
A. 1
B. 2
C. 3
D. 4
E. none of the above
4. How many terms does this expression have?
3x + xyz
A. 1
B. 2
C. 3
D. 4
E. none of the above
5. How many terms does this expression have?
3x · xyz
A. 1
B. 2
C. 3
D. 4
E. none of the above
6. How many terms does this expression have?
3x · xyz + 5x
A. 1
B. 2
C. 3
D. 4
E. none of the above
7. How many terms does this expression have?
3x · xyz · 5x
A. 1
B. 2
C. 3
D. 4
E. none of the above
c
2007
MathHands.com
ALGEBRA
Sec. 4
MathHands.com
Márquez
8. The Distributive law would say...
3x + 5x =
Solution: (3 + 5)x
9. The Distributive law would say...
9r + 5r =
Solution: (9 + 5)r
10. The Distributive law would say...
Y4+Y3 =
Solution: Y (4 + 3)
11. The Distributive law would say...
Solution:
4+
3=
x+
3=
(4 + 3)
12. The Distributive law would say...
Solution:
(x + 3)
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2007
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ALGEBRA
Sec. 4
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13. The Distributive law would say...
(x + 3 + 4) =
Solution:
x+
3+
4
14. The Distributive law would say...
(x + 3 + y) =
Solution:
x+
3+
y
15. The Distributive law would say...
x2 + 3yz + yr5 =
Solution:
x2 +
3yz +
yr5
16. The Distributive law would say...
x2 + 3yz + yr5
Solution: x2
+ 3yz
=
+ yr5
17. The Distributive law would say...
0(0 + 0) =
Solution: 0 · 0 + 0 · 0
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2007
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ALGEBRA
Sec. 4
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18. The Distributive law would say...
(3x + 5) x2 + 3yz + yr5 =
Solution: (3x + 5)x2 + (3x + 5)3yz + (3x + 5)yr5
19. The Distributive law would say...
(3 + x)5 =
Solution: NOTHING! the DL say s nothing about EXPONENTS, (and...we haven’t even learned them yet)
20. The Distributive law would say...
sin(x + 5) =
Solution: NOTHING! the DL say s nothing about functions, (and... we haven’t even learned them yet) IF sin
was a variable which represented a number, we could use DL, but it does not represent a number, it represents
a function.
21. The Distributive law would say...
1
(3 + x) 2 =
Solution: NOTHING! the DL say s nothing about EXPONENTS, (and... we haven’t even learned them yet)
22. The Distributive law would say...
(x2 + 1)4 + (x2 + 1)3 =
Solution: (x2 + 1)(4 + 3)
c
2007
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ALGEBRA
Sec. 4
MathHands.com
Márquez
23. The Distributive law would say...
5(3x) = 5 · 3 + 5 · x
A. TRUE
B. FALSE
24. The Distributive law would say...
5(3x) =
Solution: NOTHING! the DL say s nothing when all we have is ONE term..
25. The Distributive law would say...
log(x + 5) =
Solution: NOTHING! the DL say s nothing about functions, (and... we haven’t even learned them yet).. IF log
was a variable which represented a number, we could use DL, but it does not represent a number, it represents
a function.
26. The Distributive law would say...
(A + B)(C + D) =
Solution: two possible solutions...
(A + B)C + (A + B)D
OR
A(C + D) + B(C + D)
c
2007
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