Mathematics
What Are Like Terms?
3x, 3x², 3xy and 3 are four different kinds of term and none of them combine. What has to match before you can add terms, and why multiplying is allowed anyway.
Like terms
Like terms are terms whose letter parts are identical, including the powers, so they can be added or subtracted into one term.
- Also called
- Collecting like terms, Similar terms
- Where students meet it
- Grade 6 or 7 mathematics in the first lessons on simplifying expressions, and assumed in every algebraic manipulation from then on.
The short answer
Like terms have exactly the same letters raised to exactly the same powers, so they can be added or subtracted into a single term. 3x and 7x are like terms; 3x, 3x², 3xy and 3 are four different kinds and none of them combine. Only the number in front changes.
An example
4a + 3b − a + 2b = 3a + 5b
The a terms are 4a and −a, with coefficients 4 and −1, giving 3a. The b terms are 3b and 2b, giving 5b. The answer stops there: a and b are different letters, so 3a + 5b cannot be pushed together. Writing 8ab is the standard wrong answer, and it changes an addition into a multiplication.
Four terms that look related and are not
3x, 3x², 3xy and 3 all begin with a 3, and none of them can be added to any of the others. What has to match is the letter part in full: which letters appear, and to what power.
x on its own means x¹, so x and x² are as different from each other as x and y are. A term with no letters, like 3, is a constant and is only ever like another constant. Order inside a term does not matter, though — xy and yx are the same thing, so 2xy and 5yx are like terms and add to 7xy.
- 3x — one x, to the power 1
- 3x² — one x, to the power 2: a different kind of term
- 3xy — two different letters multiplied
- 3 — a constant, like only with other constants
What changes when you collect them
Only the coefficients. In 4a + 3b − a + 2b, the a terms combine to 3a and the b terms to 5b, and at no point does an a become anything other than an a. If the letter part of your answer has changed, something has gone wrong.
The sign in front of a term travels with it. Here −a is really −1a, so 4 − 1 = 3 rather than 4 − 0 = 4, which is the error you get when the invisible 1 is read as nothing at all.
Multiplying has different rules entirely
Unlike terms cannot be added, but they can always be multiplied. 3a × 5b = 15ab, and 3x × 4x = 12x². The likeness rule governs addition and subtraction only.
This is worth stating plainly because 3x and 4x behave in two ways depending on the operation: added they give 7x, multiplied they give 12x². Both are correct, and a student who has learned 'you can only combine like terms' as a blanket ban will refuse the multiplication.
Common questions
Are 3x and 3x² like terms?
No. The powers differ, and the power is part of the letter part that has to match. 3x + 3x² stays as it is — there is no simpler way to write it. If you substitute x = 2 you can see why: 6 and 12 are genuinely different quantities that happen to be built from the same letter.
Why can't 3a + 5b be simplified?
Because a and b are different quantities, and there is no way to say how many of one make the other. The tempting answer 8ab is wrong twice over: it invents a product where an addition was written, and it would give 8 × 2 × 3 = 48 for a = 2 and b = 3 when the true value is 6 + 15 = 21.
Are xy and yx like terms?
Yes, and they are actually the same term. Multiplication can be done in any order, so xy and yx are identical, which means 2xy + 5yx = 7xy. By convention the letters are written alphabetically, so yx is normally tidied to xy before anything else happens.
Can you multiply terms that are not alike?
Yes. The rule about like terms restricts adding and subtracting only. 4a × 3b = 12ab and 2x × 5x² = 10x³ are both fine — multiply the coefficients, then multiply the letters, adding the powers of any letter that appears in both.
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