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Distributing Multiplication over Addition and Subtraction Help

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By — McGraw-Hill Professional
Updated on Sep 26, 2011

Distributive Property - Multiplication Over Addition and Subtraction

Distributing multiplication over addition (and subtraction) and factoring (the opposite of distributing) are extremely important in algebra. The distributive law of multiplication over addition, a(b + c) = ab + ac , says that you can first take the sum ( b + c ) then the product ( a times the sum of b and c ) or the individual products ( ab and ac ) then the sum (the sum of ab and ac ). For instance, 12(6 + 4) could be computed as 12(6 + 4) = 12(6)+ 12(4) = 72 + 48 = 120 or as 12(6 + 4) = 12(10) = 120. The distributive law of multiplication over subtraction, a(b – c) = abac , says the same about a product and difference.

Examples

Distributing Multiplication over Addition and Subtraction Examples

 

Find practice problems and solutions at Distributing Multiplication over Addition and Subtraction Practice Problems  - Set 1.

Distributing Negative Signs (Minuses)

Sometimes you will need to “distribute” a minus sign or negative sign: –( a + b ) = – ab and –( ab ) = – a + b . You can use the distributive properties and think of –( a + b ) as (–1)( a + b ) and –( ab ) as (–1)( ab ):

–( a + b ) = (–1)( a + b ) = (–1) a +(–1) b = – a + – b = – ab

and

–( ab ) = (–1)( ab ) = (–1) a – (–1) b = – a – (–1) b

     = – a – (– b ) = – a + b .

A common mistake is to write –( a + b ) = – a + b and –( ab ) = –a –b. The minus sign and negative sign in front of the parentheses changes the signs of every term (a quantity separated by a plus or minus sign) inside the parentheses.

Examples

–(3 + x ) = –3 – x

–( yx 2 ) = – y + x 2

–(–2 + y ) = 2 – y   

–(–9 – y ) = 9 + y

–(2 + x – 3 y ) = –2 – x + 3 y    

–( x 2x – 2) = – x 2 + x + 2

–(–4 x – 7 y – 2) = 4 x + 7 y + 2

Find practice problems and solutions at Distributing Multiplication over Addition and Subtraction Practice Problems - Set 2.

Distributing Negative Quantities

Distributing negative quantities has the same effect on signs as distributing a minus sign: every sign in the parentheses changes.

Examples

–8(4 + 5 x ) = –32 – 40 x

xy (1 – x ) = – xy + x 2 y

–3 x 2 (–2 y + 9 x ) = 6 x 2 y – 27 x 3

–100(–4 – x ) = 400 + 100 x

Find practice problems and solutions at Distributing Multiplication over Addition and Subtraction Practice Problems - Set 3.

More practice problems for this concept can be found at: Algebra Factoring Practice Test.

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