Distributive Property
The distributive property multiplies the number outside parentheses by each term inside: . In , the multiplies the and it multiplies the , giving . Both terms get the multiplication — that is the whole property.
You need this move because cannot be combined into one term, so the multiplication cannot wait for the parentheses to finish. Distributing is how algebra clears parentheses, and it shows up in nearly every equation you will solve after this unit.
Multiply each term inside
Draw two arrows in your head — one from the outside number to each inside term. For : and , so .
The most common error is multiplying only the first term: writing . Check yourself by counting terms — two terms inside the parentheses means two multiplications, every time.
Subtraction inside the parentheses
Treat the subtraction as a negative term riding along. means and , which gives . The minus sign inside survives into the answer.
A negative number outside
When the outside number is negative, it multiplies each term — sign included. gives and , so the answer is .
And when negative meets negative, the product is positive: . Every sign inside the parentheses gets flipped by a negative distributor. If exactly one sign flipped and the other did not, go back and find the error.
Worked examples
Example 1: a positive number outside
Distribute .
Answer:
Example 2: subtraction inside
Distribute .
Answer:
Example 3: a negative number outside
Distribute .
Answer:
Try one yourself
Common questions
Why can't I just add what's inside the parentheses first?
If the inside is all numbers, you can — works fine. But cannot be added into a single term, so distributing is the only way to clear the parentheses.
What happens with a negative outside and a minus inside?
The two negatives multiply to a positive. . Every term inside gets its sign flipped by the negative distributor — if only one term flipped, there is a sign error.
What if there are three terms inside the parentheses?
Distribute to all three. . The number outside multiplies every term inside, no matter how many there are.
Does distributing work with a variable outside?
Yes, the same way. . The outside factor multiplies each term inside, whether it is a number, a variable, or both.
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