The distributive property: why the outside factor reaches every term
Distribution is multiplication across a sum. It expands an expression without changing its value—and it works in reverse when factoring.
LaTeX article Updated July 13, 2026
Why distribution is legal
Imagine b + c as a length split into two pieces. A rectangle of height a has total area a(b + c), which is also the sum of the two smaller areas ab and ac.
The same equality holds for negative numbers, variables, and expressions. Distribution is structural, not a visual trick.
The negative sign is a factor
A minus sign before parentheses can be read as multiplication by negative one. Every term inside must change sign.
This is the source of many equation errors: the first term changes, but the later terms are accidentally left alone.
Factoring runs the movie backward
If two terms share a factor, pull it outside and leave behind what remains after division. Expanding can verify the result immediately.
The greatest common factor produces the most useful factored form because nothing common remains inside.
Worked example
Common mistakes
- Multiplying only the first term in parentheses.
- Forgetting that a leading minus means −1.
- Changing signs without multiplying magnitudes.
Keep these ideas
- Every term receives the outside factor.
- A minus sign can be treated as −1.
- Factoring is reverse distribution.