 More About the Distributive Property MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts When the distributive property is mentioned, we think of:
a ( b + c ) = a • b + a • c = ab + ac

To be more specific, it is officially called:
"the distributive property of multiplication over addition"

Note: you can distribute "over" addition (or subtraction, which is adding a negative),
but you cannot distribute "over" multiplication or division. There are some really interesting ideas associated with the distributive property. Don't turn the channel just yet. You may find these ideas interesting and useful! Numerical Connections:

 1. Distributive Property versus Order of Operations When dealing with numerical values and simplifying a ( b + c ), the "distributive property" and the "order of operations", are each capable of accomplishing the same task. Distributive Property: 3 ( 4 + 2 ) = 3 • 4 + 3 • 2          = 12 + 6   = 18 Order of Operations: 3 ( 4 + 2 ) = 3 ( 6 )             = 18 2. Distributive Property making Multiplication Easier When working without a calculator, you can use the distributive property to make your multiplication computations easier by breaking a number into components that are easily multiplied in your head. Multiply: 6 x 148 Expressing as addition. 6 ( 100 + 40 + 8 ) = 600 + 240 + 48          = 840 + 48   = 888 Multiply: 8 x 19 Expressing as subtraction. 8 ( 20 - 1 ) = 160 - 8 = 152 Algebraic Connections:

 3. Distributive Property over Multiple Terms The distributive property works as multiplication over addition "of any number of terms". While we think of a ( b + c ) = a • b + a • c, we could just as easily think of a ( b + c + d + e) = a • b + a • c + a • d + a • e (or any number of terms inside the parentheses). Distributive Property: 3 ( 4 + 2 + 6 + 1 ) = 3 • 4 + 3 • 2 + 3 • 6 + 3 • 1          = 12 + 6 + 18 + 3   = 39 Distributive Property: 2 ( 4n + 1 + 2n + 7 + n) = 2 • 4n + 2 • 1 + 2 • 2n + 2 • 7 + 2 • n = 8n + 2 + 4n + 14 + 2n             = 14n + 16 This concept is particularly helpful in Algebra as it lets us distribute a factor over a polynomial. 4. The Distributive Property in Reverse You can work the distributive property "backward" so that a • b + a • c = a ( b + c ). While is can be referred to as reversing the distributive property, it is actually just pulling out the greatest common factor. Distributive Property: 5a + 5b = 5 • a + 5 • b = 5( a + b ) common factor of 5 Distributive Property: 3x2 + 6x + 3 = 3 • x2 + 3 • 2x + 3 • 1 = 3( x2 + 2x + 1 ) common factor of 3 5. The Distributive Property with Two Sums When two sums are involved, the distributive property is actually being applied twice. It does not matter which parentheses is first multiplied out. Think of one of the parentheses are being a single entity. Distributive Property: (a + b) • (a - b) = a • (a - b) + b• (a - b) = a2 - ab + ba - b2 = a2 - b2 Distributive Property: (a + b) • (a - b) = (a + b) • a - (a + b) • b = a2 + ba - ab - b2 = a2 - b2 6 The Distributive Property in Geometry When working in geometry, a visual interpretation of the distributive property at work while finding the area of rectangles (using positive numbers for the variables) is shown below.  7.
We know that there is a Distributive Property of Multiplication over Addition.
But is there a Distributive Property of "Division" Over Addition?

No. Not really.
One of the parts of the definition of "distributive property" (or just "distributive") is that the operation is both left-distributive and right-distributive. (sources: WolframAlpha, Wikipedia)
This is true for "
multiplication over addition" because multiplication is commutative.
 Left distributive a • ( b + c ) = a • b + a • c TRUE Right distributive ( b + c ) • a = b • a + c • a TRUE

Since division is NOT commutative, "
division over addition" fails to be left-distributive.
 Left distributive also written as: FALSE Right distributive also written as: TRUE

Since "division over addition" fails this portion of the definition of the term "distributive" there is no "distributive property of division over addition". Perhaps it could (or should if used) be called the "right-distributive property of division over addition".

NOTE: The right distributive trait of division in this situation, , can show the desired result as .

This same result can be obtained using the "distributive property of multiplication over division" where "a" in replaced by the fraction 1/a. or Note: When using a fraction, the numerator does not have to be 1.  