Adding and Subtracting Polynomials
A polynomial is a sum of terms like , , and . Adding or subtracting two of them does not require any new rule. You are just collecting the pieces that match, the same way you would put all the quarters in one pile and all the dimes in another.
Terms match when they have the same variable raised to the same exponent. So and combine into , but and stay separate. Everything in this lesson comes from that one idea, plus one sign rule for subtraction.
Like terms are the whole game
Like terms have the identical variable part. and are like terms because both carry . and are not, because the exponents differ. Plain numbers such as and are like terms with each other.
To combine like terms, add the coefficients and leave the variable part alone. , not . The exponent counts how many factors a term has, and adding more of the same term never changes that count.
Adding: drop the parentheses and collect
When the two polynomials are being added, the parentheses do no work. Every sign inside stays exactly as it is, so you can erase the parentheses and rewrite the whole thing as one long sum.
Then sweep through and gather the terms, the terms, and the numbers. Write the result in standard form, meaning highest exponent first and the plain number last. If a power appears in only one of the polynomials, it simply carries down unchanged.
Subtracting: every sign in the second polynomial flips
Subtraction is where the parentheses matter. The minus sign in front applies to the entire second polynomial, not just its first term. Distribute it: each inside becomes and each inside becomes . After that you are adding again, and the rest is the same collecting work.
This is the error worth guarding against. In , students often flip the and the but forget the , which should become . Rewrite the flipped version on its own line before combining anything, and check that you changed the sign of every single term.
Worked examples
Example 1: adding two trinomials
Add .
Answer:
Example 2: a missing power
Add .
Answer:
Example 3: subtracting
Subtract .
Answer:
Example 4: subtracting when a power is missing
Subtract .
Answer:
Try one yourself
Common questions
What makes two terms like terms?
Same variable, same exponent. and are like terms. and are not, and neither are and . Only like terms can be combined into a single term.
Do the exponents change when I add polynomials?
No. You add the coefficients and keep the variable part as it is, so . Getting means the exponents were added by mistake, which is a multiplication rule, not an addition rule.
Why does subtraction change so many signs?
Because the minus sign belongs to the whole polynomial in parentheses, not just its first term. Subtracting means subtracting , subtracting , and subtracting , and subtracting a negative adds. Rewriting it as before you combine keeps this straight.
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