**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)* = *ab* – *ac* , says the same about a product and difference.

**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* ) = – *a* – *b* and –( *a* – *b* ) = – *a* + *b* . You can use the distributive properties and think of –( *a* + *b* ) as (–1)( *a* + *b* ) and –( *a* – *b* ) as (–1)( *a* – *b* ):

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

and

–( *a* – *b* ) = (–1)( *a* – *b* ) = (–1) *a* – (–1) *b* = – *a* – (–1) *b*

= – *a* – (– *b* ) = – *a* + *b* .

A common mistake is to write –( *a* + *b* ) = – *a* + *b* and –( *a* – *b* ) = –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*

–( *y* – *x* ^{2} ) = – *y* + *x* ^{2}

–(–2 + *y* ) = 2 – *y *

* *–(–9 – *y* ) = 9 + *y*

–(2 + *x* – 3 *y* ) = –2 – *x* + 3 *y *

* *–( *x* ^{2} – *x* – 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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