What are like terms?
Like terms contain the same variables raised to the same powers. In 5x², −3x², and x², the variable part x² matches, so the terms are like. Their coefficients are 5, −3, and 1. Constants such as 8 and −2 are also like because they have no variable part.
Terms such as 4x² and 4x are unlike because their exponents differ. Likewise, x and y are unlike, and x²y is different from xy². The coefficient does not determine whether terms match; the complete variable part does.
How to combine like terms step by step
The commutative and associative properties allow terms joined by addition to be reordered and regrouped. Once each subtraction sign is attached to the term that follows it, matching groups can be collected safely.
- Separate the expression into terms, keeping each plus or minus sign attached.
- Identify the variable part and exponent of every term.
- Group terms with identical variable parts; group constants together.
- Add or subtract only the numerical coefficients in each group.
- Keep the shared variable part unchanged and write the result in descending-power order.
- Compare the original and final polynomial coefficients to verify equivalence.
Worked example: 4x² + 3x − x² + 8 − 5x
Group matching powers: (4x² − x²) + (3x − 5x) + 8. For the quadratic terms, 4 − 1 = 3, so they become 3x². For the linear terms, 3 − 5 = −2, so they become −2x. The constant remains 8.
The simplified polynomial is 3x² − 2x + 8. Notice that x² and x stayed separate. If x = 2, the original and simplified forms both evaluate to 16, giving a quick numerical check in addition to the symbolic coefficient check.
Distribute before collecting hidden like terms
Some like terms are hidden inside parentheses. In 3(x + 2) − 2x, distribute first to get 3x + 6 − 2x. Now the x-terms are visible and combine to x, giving x + 6.
A negative sign before a group also distributes. For 5x − (2x − 4), rewrite the subtracted group as −2x + 4, then combine to obtain 3x + 4. Skipping this sign change creates the wrong constant.
Coefficients, powers, and zero groups
A missing written coefficient is 1, and a leading minus without a number means −1. If a coefficient group sums to zero, that whole term disappears: 6x³ − 6x³ = 0. A missing power in the final polynomial therefore has coefficient zero.
Combining terms uses addition, so the exponent stays unchanged. Exponents are added only when multiplying powers with the same base. Keeping these two rules separate prevents mistakes such as turning x² + x² into x⁴.
Common mistakes when collecting terms
Do not combine based only on a shared letter if the exponents differ. Do not ignore negative coefficients, and do not change the variable power after adding coefficients. Constants can combine with constants but not with variable terms.
When checking your work, list the coefficient for every power before and after. Equivalent polynomials have matching coefficients across the supported domain, even if their terms were originally arranged differently.