11.7 Distributive Property
The distributive property allows multiplication to be distributed over addition, simplifying expressions and forming the foundation of algebraic manipulation.
Distributive Property describes how multiplying a sum or difference by a factor produces the same result as multiplying each term of that sum or difference by the factor individually and then combining the resulting products, linking the operations of multiplication and addition together in a way no single one of the other properties does alone.
Distribution over Addition and Subtraction
Multiplication over Addition
The distributive property applied to addition states that a factor multiplying a sum can instead be multiplied by each addend separately, with the resulting products then added together, producing the identical value as multiplying the original sum directly.
Multiplication over Subtraction
The distributive property applied to subtraction states that a factor multiplying a difference can instead be multiplied by each of the two terms separately, with the resulting products then subtracted in the same order, producing the identical value as multiplying the original difference directly.
Which Side the Factor Distributes From
Left-Side Distribution
When the multiplying factor is written before the grouped sum or difference, it distributes from the left, multiplying each term inside the group while remaining in front of each resulting product.
Right-Side Distribution
When the multiplying factor is written after the grouped sum or difference, it distributes from the right, multiplying each term inside the group while remaining behind each resulting product, producing the same result as left-side distribution by the commutative property of multiplication.
Distributing across Every Term
Distribution to Every Grouped Term
When the grouped expression contains more than two terms, the factor distributes to every single one of them individually, with no term inside the group left unmultiplied, no matter how many terms the group contains.
Distribution with Different Kinds of Factors
Distribution with a Positive Factor
When the distributed factor is positive, each term inside the group keeps its original sign after multiplication, since multiplying by a positive number never flips the sign of the quantity it multiplies.
Distribution with a Negative Factor
When the distributed factor is negative, every term inside the group has its sign reversed after multiplication, since multiplying each term by a negative number flips its sign, including changing what was a subtraction inside the group into an addition, and vice versa.
Distribution with a Fractional Factor
When the distributed factor is a fraction, each term inside the group is multiplied by that same fraction individually, following the ordinary rules of fraction multiplication for each resulting product.
Distribution and Factoring as Reverse Processes
Distributed and Factored Form Equivalence
An expression written as a factor multiplying a grouped sum and the same expression written after distribution as a sum of separate products always represent the identical value; the two forms are equivalent, differing only in whether the common factor is written once outside a group or repeated separately with each term.
Confirming a Distribution
Numerical Verification of Distribution
A completed distribution can be checked by substituting a chosen numerical value for any variables involved into both the original grouped form and the distributed form separately, and confirming that both forms evaluate to the same number; a mismatch indicates that a term was skipped or a sign was applied incorrectly during the distribution.