How do you combine like terms?
Find the terms that share a variable and power, add their coefficients, and keep the variable part as it is. 5x + 2x is 7x because you add 5 and 2, while the x stays put.
Combining like terms works because of the distributive property: 5x + 2x = (5 + 2)x. You're adding counts of the same thing, the way 5 apples plus 2 apples is 7 apples.
- Underline or circle each group of like terms. Take the sign in front of each term with it.
- Add the coefficients within each group.
- Write the groups in standard form, highest power first.
Worked example: 4x² + 3x − x² + 8 − 5x
The answer is 3x² − 2x + 8. There are three groups here: x² terms, x terms, and a constant.
Notice that −x² has a hidden coefficient of −1, so 4 − 1 = 3. The 3x and −5x give 3 − 5 = −2. The 8 has nothing to pair with, so it stays.
Why are x² and x not like terms?
They have the same letter but different powers, so they measure different things. Think of x as a length and x² as an area: adding a length to an area doesn't give you more of either one.
Try x = 3. Then x² + x = 9 + 3 = 12. If you wrongly wrote 2x², you'd get 18, and 2x would give 6. Neither matches, which is why x² + x is already as simple as it gets.
| Pair | Like terms? | Why |
|---|---|---|
| 7x and −2x | Yes | Same variable, same power |
| 4x² and x² | Yes | Both are x² |
| 3x² and 3x | No | Different powers |
| 6 and −11 | Yes | Both are constants |
| 5x and 5 | No | One has a variable, one doesn't |
Are constants like terms?
Yes. Every plain number is a like term with every other plain number, so 8 − 3 + 1 combines to 6. A constant never combines with a term that has a variable in it.
In 6 − 2(x − 1), the constants only appear after you distribute: 6 − 2x + 2. Then 6 + 2 = 8, and the answer is −2x + 8. Clear the brackets first or you'll miss constants hiding inside.
How do you combine like terms with fraction coefficients?
Add the fractions the usual way, with a common denominator, and keep the variable. (1/2)x + (1/3)x becomes (3/6 + 2/6)x, which is (5/6)x.
If the denominators already match, just add the tops: (3/4)x − (1/4)x = (2/4)x = (1/2)x. When you type these into the calculator, put the fraction in brackets, like (3/4)x, so it's clear which part is the coefficient.
What sign mistakes do students make when combining like terms?
The usual mistake is dropping the minus sign in front of a term. A term owns the sign directly before it, so in 3x − 5x the second term is −5x, not 5x.
A second trap is forgetting the invisible 1. On its own, x means 1x, and −x means −1x. So x² + 3x − 2x² + x has x² coefficients 1 and −2 (giving −x²) and x coefficients 3 and 1 (giving 4x). The result is −x² + 4x.
Third, watch a minus sign in front of brackets. In 5x − (2x − 3), the minus applies to both terms inside, giving 5x − 2x + 3 = 3x + 3.
How do you combine like terms when there are brackets?
Distribute first, then combine. Like terms often sit inside different brackets, and you can only see them once the brackets are gone.
Take 2(x² + 3x) − 3(x² − x + 1). The −3 multiplies all three terms in the second bracket, which flips each sign.
A leading negative like −x² is fine to leave at the front. Standard form cares about the order of powers, not about the first sign being positive.
What can't this calculator combine?
It works with one variable at a time, so expressions mixing x and y, such as 3xy + 2x, aren't supported. It also doesn't accept a variable in a denominator.
Within those limits, it handles powers up to 6, fraction coefficients, and brackets. If your expression has brackets, the simplify calculator shows the distributing step in more detail.