Distributing Polynomials
Distributing means multiplying one term by everything inside a set of parentheses. In , the multiplies both the and the , giving . You have seen this with plain numbers — now the term out front can have a variable too, like .
Two rules carry the whole topic: multiply the coefficients, and add the exponents on matching variables. So , because . Once those two moves are automatic, distributing any monomial across any polynomial is the same short routine.
Distribute to every term
The term in front must reach every term inside the parentheses — no exceptions. For that means three separate products: , then , then . The answer is .
For each product, handle the numbers and the variables separately: multiply the coefficients, then add the exponents on the variable. In , the coefficients give and the variables give , so the product is .
The most common mistake is multiplying only the first term inside and forgetting the rest. Draw arrows from the front term to each inside term if it helps — every arrow is one product.
Distributing a negative
When the front term is negative, its sign travels with it into every product. In : , then , then . Every sign in the answer changes from the original: .
Slow down on the middle terms. A negative times a negative is positive, and that flip is exactly where rushed work goes wrong.
Distribute first, then combine like terms
Many problems mix distribution with extra terms, like . Distribute first to get , then combine the like terms: . The simplified answer is .
The order matters — you cannot combine anything with a term that is still locked inside parentheses.
Worked examples
Example 1: a monomial times a trinomial
Multiply .
Answer:
Example 2: a negative front term
Multiply .
Answer:
Example 3: distribute, then combine like terms
Simplify .
Answer:
Try one yourself
Common questions
Why does equal and not ?
Because means , and when you multiply matching bases you add the exponents: . The coefficient just rides along.
Do I add or multiply the coefficients?
Multiply them — this is multiplication, not combining like terms. In , the coefficients give and the variables give , so the product is .
What if there are two variables, like ?
Same rules, one variable at a time. and , so the answer is . Exponents only add on matching letters.
How can I check a distribution?
Factor your answer back: pull the front term out of every term of your result and you should land exactly on the original parentheses. For instance, confirms Example-style work.
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