As you write expressions, it helps to have names for their parts, so we can talk about them clearly.
Key idea
Terms are the parts that are added (or subtracted). A coefficient is the number multiplying a variable. A constant is a term with no variable. A factor is something that is multiplied.
| In | $2(x + 3) + 4y + 7$ |
|---|---|
| terms | $2(x + 3)$, $4y$ and $7$: three terms |
| coefficient of $y$ | $4$ |
| constant | $7$ |
| factors of the first term | $2$ and $(x + 3)$ |
Why it works
Parentheses group a sum into a single quantity. In $2(x + 3)$, the 2 multiplies the whole sum, so $2(x + 3)$ is one term: a product of the two factors 2 and $(x + 3)$. The + sign inside the parentheses is part of that factor, so it does not split the expression into more terms. A letter written alone, like $y$, has coefficient 1, because $y = 1 \cdot y$.
Worked example: Count and name
In $5m + n + 9$, how many terms are there, and what is the coefficient of $n$?
- The terms are $5m$, $n$ and $9$: three terms.
- $n$ is $1 \cdot n$, so its coefficient is 1.
Answer: 3 terms; the coefficient of $n$ is 1.
Worked example: Parentheses make one term
How many terms are in $6(a + 2) + 3a$?
- $6(a + 2)$ is one product, one term.
- $3a$ is the other term.
Answer: 2 terms
Worked example: Describe in words
Describe $8(k + 1)$ in words.
- It is a product of two factors.
- The second factor is itself a sum of two terms, $k$ and 1.
Answer: The product of 8 and the sum of $k$ and 1.
Common mistake
Counting the + inside parentheses as a new term, so that $6(a + 2) + 3a$ seems to have 3 terms. Quick check: only + and − signs that are outside every parentheses separate terms. Another slip is saying $n$ has no coefficient; it has coefficient 1.