Math101 · Independent Ontario learning library
Math101learn.math101.caDistributive Property
The distributive property connects multiplication with addition and subtraction, allowing brackets to be expanded or common factors to be revealed.
Multiplying a sum or difference means multiplying every term inside the grouping: $a(b+c)=ab+ac$.
The area idea
A rectangle with height $a$ and total width $b+c$ has area $a(b+c)$. Split it into widths $b$ and $c$; the two smaller areas are $ab$ and $ac$. Because the same region is measured both ways,
This visual argument explains why no term inside the brackets can be skipped.
Expanding brackets
To expand, multiply the outside factor by each term inside:
The minus sign belongs to $-3$. Treating subtraction as addition of a negative makes distribution consistent.
Negative factors
A negative sign before brackets is a factor of $-1$:
Every sign inside changes because every term is multiplied by $-1$. With a larger negative factor, multiply coefficients and apply the usual sign rules.
Combining after distribution
Distribution often reveals like terms:
Work in two stages: expand every bracket, then combine like terms. Trying to do both mentally makes sign errors harder to locate.
Factoring is reverse distribution
The property works in both directions:
Expanding multiplies a common factor into every term. Factoring identifies that common factor and moves it outside. Multiplying back is a quick verification.
Efficient mental arithmetic
Distribution is useful without variables. For example,
Choosing a nearby friendly number turns a difficult product into two manageable products.
In equations
When solving $3(x+2)=21$, you may distribute to get $3x+6=21$, or divide both sides by $3$ first to get $x+2=7$. Both methods preserve equality. Choose the route with fewer risky steps.
With brackets on both sides, expand and simplify each side before moving variable terms.
Common mistakes
Multiplying only the first term. $5(x+2)$ is $5x+10$, not $5x+2$.
Losing the sign of a negative factor. Write the intermediate products when needed.
Distributing an exponent over addition. $(x+2)^2$ is not $x^2+4$; it means $(x+2)(x+2)$.
Combining unlike terms after expanding. $3x+6$ cannot become $9x$.
Quick self-check
- What is the complete outside factor, including its sign?
- Did it multiply every term inside the grouping?
- Have I combined only genuine like terms?
- Can I factor or substitute to check the result?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Expand −3(2x + 4 − y).
- −3(2x) = −6x
- −3(4) = −12
- −3(−y) = +3y
End of lesson
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