To collect like terms, add or subtract only the terms that have exactly the same letters with exactly the same powers, and leave those powers unchanged. The number in front (the coefficient) is the only thing that changes. This appears in almost every algebra question, often as the first tidy-up step before expanding, factorising or solving.
It builds on indices and powers and leads into the rest of algebraic structure.
What makes two terms “like”?
Two terms are like terms when their letter parts match completely. 4x² and 7x² are like terms, because both have x². 4x² and 7x are not, because one has x² and the other has x.
Think of it as counting the same object. Four x² plus seven x² gives eleven x², just as four mangoes plus seven mangoes gives eleven mangoes. Counting more does not change the object, so the power stays put.
How to collect terms, step by step
- Write every term with its sign. The sign belongs to the term that follows it.
- Sort by letter part. Group terms with identical letters and powers, for example all the x²y terms together.
- Combine the coefficients of each group, using signed-number arithmetic.
- Keep the letter part exactly as it was.
- Write the answer tidily, usually with the highest powers first, and keep any constant term.
Worked example
Simplify 3x²y − 5xy² + 2x²y + xy² − 4
Step 1, list terms with signs: +3x²y, −5xy², +2x²y, +xy², −4.
Step 2, sort: the x²y terms are +3x²y and +2x²y. The xy² terms are −5xy² and +xy². The constant is −4.
Step 3, combine coefficients: 3 + 2 = 5, so 5x²y. For the others, −5 + 1 = −4, so −4xy². Remember that xy² means 1xy².
Step 4, answer:
5x²y − 4xy² − 4
Check: put x = 2 and y = 1 into the original. 3(4)(1) = 12, −5(2)(1) = −10, 2(4)(1) = 8, (2)(1) = 2, then −4. The total is 12 − 10 + 8 + 2 − 4 = 8. In the answer, 5(4)(1) − 4(2)(1) − 4 = 20 − 8 − 4 = 8. They match.
The mistake to watch for
A common slip is to add the powers as well as the coefficients, or to treat similar-looking terms as the same.
Mistaken answer: 4a² + 3a² = 7a⁴
The student added the coefficients correctly (4 + 3 = 7) but also added the powers (2 + 2 = 4). Adding powers belongs to multiplication, not to collecting terms.
The correction is to ask “how many of the same object?”. Four a² plus three a² is seven a², so 7a².
A second version of the same slip is writing 2x²y + xy² as 3x³y³. Those two terms are not like terms, so the expression 2x²y + xy² cannot be shortened.
A substitution check exposes both errors quickly. With a = 2, 4a² + 3a² is 16 + 12 = 28, and 7a² is 28, but 7a⁴ is 112.
Check yourself
Try these without a calculator, then open each answer.
1. Simplify 6a²b − a²b + 3ab²
Show answer
The a²b terms: 6 − 1 = 5, giving 5a²b. The ab² term is alone, so it stays as 3ab².
5a²b + 3ab²
Check with a = 1 and b = 2: the original is 12 − 2 + 12 = 22, and 5(2) + 3(4) = 22.
2. Simplify 7p − 2q² + 3p² − 7p + 5q²
Show answer
The p terms cancel: 7p − 7p = 0. The q² terms: −2 + 5 = 3, giving 3q². The p² term stays.
3p² + 3q²
Check with p = 2 and q = 1: the original is 14 − 2 + 12 − 14 + 5 = 15, and 12 + 3 = 15.
3. A student writes 3x² + 2x = 5x³. Show with x = 2 that this is wrong.
Show answer
When x = 2, 3x² + 2x = 12 + 4 = 16. But 5x³ = 5 × 8 = 40. The two sides differ, so the statement is false. The terms are not like terms, so 3x² + 2x cannot be simplified.
Where this leads next
Once collecting terms is automatic, move on to expanding a product containing negative terms, where you will collect terms again after removing brackets. The non-calculator working trainer is useful for checking the coefficient arithmetic when numbers get awkward.
If you understand each step here but still slip when a question is long, that pattern is what our teachers look for in online one-to-one Mathematics tuition.