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The Distributive Property

is used to

simplify expressions containing parentheses



Math - Distributive Property

Distributive Property -


Properties of Real Numbers:



Real numbers have unique properties

which make them particularly useful in everyday life.

First , Real numbers are an ordered set of numbers. This means real numbers are sequential. The numerical value of every real number fits between the numerical values two other real numbers.

Everyone is familiar with this idea since all measurements (weight, the purchasing power of money, the speed of a car, etc.) depend upon the fact that some numbers have a higher value than other numbers. Ten is greater than five, and five is greater than four . . . and so on.

Second , we never run out of real numbers. The quantity of real numbers available is not fixed. There are an infinite number of values available. The availability of numbers expands without end. Real numbers are not simply a finite "row of separate points" on a number line. There is always another real number whose value falls between any two real numbers (this is called the "density" property).

Third , when real numbers are added or multiplied, the result is always another real number (this is called the "closure" property). [This is not the case with all arithmetic operations. For example, the square root of a -1 yields an imaginary number.]

With these three points in mind, the question is: How can we use real numbers in practical calculations? What rules apply?

  • How should numbers be added, subtracted, multiplied, and divided? What latitude do we have?


  • Does it matter what we do first? second? third? . . .


  • Can we add a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?


  • Can we multiply a series of numbers together in any order? Will the final answer be the same regardless of the order we choose?

The following properties of real numbers answers these types of questions. The property characteristics which follow show how much latitude you have to change the mechanics of calculations which use real numbers without changing the results.

  • Associative Property


  • Commutative Property


  • Distributive Property


  • Identity Property


  • Inverse Property



The Seven Fundamental Properties of Real Numbers - click description





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Math - Distributive Property Distributive Property . . .

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The Distributive Property combines the following two operations in mathematics: multiplication and addition . It is formally defined as "the distribution of multiplication over addition".

There is only one Distributive Property which combines (both) addition and multiplication. There is not a separate Distributive Property for addition, and another Distributive Property for multiplication.

The Distributive Property allows us to simplify an expression containing parentheses to an expression without parentheses.

As an example , the Distributive Property can be used to multiply the following sum of two numbers by 3 :



5 + 6


This problem would ordinarily be written as expression which contains parentheses:

3 * ( 5 + 6 )


To evaluate the expression, you could add the 5 and the 6 together first, and then multiply by 3


5 + 6 = 11


3 * 11 = 33


However, you can also multiply each of the two numbers by 3 separately, and then add the results together.



3 * 5 = 15


3 * 6 = 18


15 + 18 = 33


The results are the same either way.



Simply put, the Distributive Property states that it makes no difference whether you:



- multiply 3 times each number individually, then add the result

or

- add the two numbers together, then multiply the result by 3


The Distributive Law states that the following expressions are equivalent to one another:


3 * ( 5 + 6 ) = 3 * 5 + 3 * 6 = 33



Mental Math - the Distributive Law simplifies calculations


Knowing that the two expressions shown above are equivalent is very useful when multiplying numbers using mental math.


For example, multiply the following two numbers using mental math:


5 * 723


Using the Distributive Law, this number 723 can be arbitrarily spit into parts, and re-written as shown below. This is called Decomposing the Multiplier .


5 * 723 = 5 * ( 700 + 20 + 3 )



The multiplication process can be completed as follows:



5 * ( 700 + 20 + 3 ) = 5 * 700 + 5 * 20 + 5 * 3


= 3500 + 100 + 15


= 3615



Algebra - the Distributive Law helps simplify algebraic expressions

The Distributive Law is even more important in Algebra. The values of the variables used in Algebra are unknown. The Distributive Law allows you to simplify an algebraic expression without knowing the actual values represented by the variables.


The Distributive Law can be stated using algebraic notation as follows:



a * ( b + c ) = a * b + a * c



If more than three variables are used, the Distributive Law can be stated:



a * ( b + c + d + e + ... + z ) = a * b + a * c + a * d + a * e + ... + a * z




Example Problems in Algebra using the Distributive Law :



Example Problems Using the Horizontal Format :



Example 1:


3 ( x + 9 ) = 3 * ( x + 9 )


= 3 * x + 3 * 9


= 3x + 27


Example 2:


r ( x + 9 ) = r * ( x + 9 )



= r * x + r * 9



= rx + 9r



Example Problems Using a Multiplication Table :


Example 1:


3 ( x + 9 )


(+3) ( + x + 9 )






+x
+9
+3
(+3) * (+x)
(+3) * (+9)

Multiplication Table



(+3) ( + x + 9 )



= [ (+3) * (+x) ] + [ (+3) * (+9) ]



= 3x + 27



Example 2:


( x + 2 ) ( x - 5 )



( + x + 2 ) ( + x - 5 )






+x
-5

+x
(+x) * (+x)
(+x) * (-5)
+2
(+2) * (+x)
(+2) * (-5)

Multiplication Table



= [ x * x ] - [ x * 5 ] + [ 2 * x ] - [ 2 * 5 ]


= x2 - 5x + 2x - 10


= x2 + [- 5x + 2x] - 10


= x2 - 3x - 10



Example Problems Using the Vertical Format :


Example 1:



( x + 2 ) ( x - 5 )


( + x + 2 ) ( + x - 5 )


+ x −   5 

x    + x +   2 

2 x −  10 Math - arrow left Math - arrow left Math - arrow left (+2) * ( + x - 5 )

x2 5 x  Math - arrow left Math - arrow left Math - arrow left Math - arrow left (+x) * ( + x - 5 )

_____________

x2 3 x − 10  

Example 2:



( x + 2 ) ( x2 + x - 5 )


( + x + 2 ) ( + x2 +  x  −  5 )



+ x2      +    x  −    5

x                  x   +   

+ 2 x2 +    2 x  − 1 0 Math - arrow left Math - arrow left Math - arrow left (+2) * ( + x2 +  x  −  5 )

+ x3 +    x2  −  5 x Math - arrow left Math - arrow left Math - arrow left Math - arrow left (+x) * ( + x2 +  x  −  5 )

_______________________

x3  +  3 x2   −  3 x  − 1 0