Distributive Property (cont)
Expressions like need two moves, in a fixed order: distribute first, then combine like terms. Distributing clears the parentheses, and once everything is out in the open, the like terms collapse together the usual way.
The order is not negotiable. The in cannot touch the while the is still locked inside the parentheses with a waiting to multiply it. Distribute, and only then combine.
The two-move routine
Move 1 — distribute: multiply the outside number by each term inside the parentheses. becomes .
Move 2 — combine like terms: , so the answer is . The variable term had nothing to pair with, so it rides through unchanged.
A minus sign in front of the parentheses
In , the number being distributed is — the minus sign belongs to it. Distributing gives . Notice the last term became , because .
Then combine: , so the answer is . The classic mistake is subtracting first to get — but multiplication binds before subtraction, so the must distribute before anything is subtracted.
A lone variable joins in
In , the out front just waits its turn. Distribute: . Combine, remembering the lone has a coefficient of : . The answer is .
Worked examples
Example 1: distribute, then combine constants
Simplify .
Answer:
Example 2: a minus sign distributes
Simplify .
Answer:
Example 3: a lone variable joins in
Simplify .
Answer:
Try one yourself
Common questions
In , why can't I do first?
Order of operations — multiplication comes before subtraction. The is attached to the parentheses by multiplication, so it must distribute first. The waits until the parentheses are gone.
Do I distribute the minus sign too?
Yes. In you distribute , not . Both inside terms feel it: and . Distributing and then subtracting the whole result also works, but treating the number and its sign as one package causes fewer errors.
How do I know when I'm done simplifying?
No parentheses remain, and no two remaining terms are like terms. is done — a variable term and a constant cannot combine. If you can still spot a matching pair, keep going.
What about a bare minus in front, like ?
A bare minus means a coefficient of . Distribute it: . Every sign inside gets flipped.
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