Parts of an Expression: Terms, Coefficients and Constants

Terms split at the plus and minus signs

$$4x + 7y - 3$$

That expression has three terms:

Term	Type
$4x$	variable term
$7y$	variable term
$-3$	constant

Plus and minus signs are where an expression comes apart. Everything between two of them is one term.

The sign travels with the term it precedes. In $5x - 3$ the second term is $-3$, not $3$, because subtracting $3$ is the same as adding $-3$.

Inside a term

$$4x \quad\Rightarrow\quad \underbrace{4}_{\text{coefficient}} \times \underbrace{x}_{\text{variable}}$$

The coefficient is the number multiplying the variable. The pieces being multiplied together — here $4$ and $x$ — are the factors of the term.

A term with no variable is a constant. Its value is fixed no matter what the variables do, which is where the name comes from.

Expression	Terms	Coefficients	Constant
$3x + 8$	$3x$, $8$	$3$	$8$
$x - 5$	$x$, $-5$	$1$	$-5$
$2a + 7b - 1$	$2a$, $7b$, $-1$	$2$, $7$	$-1$
$-6m$	$-6m$	$-6$	none

Naming a whole expression

An expression is named after its last operation — the one performed after everything else.

Expression	Name	Because
$3x + 5$	a sum	the final step adds
$3x - 5$	a difference	the final step subtracts
$3(x + 5)$	a product	the bracket is formed, then multiplied
$\tfrac{x + 5}{3}$	a quotient	the sum is formed, then divided

$3x + 5$ and $3(x + 5)$ contain the same symbols and are named differently, because the bracket changes which operation comes last.

That is worth noticing precisely because the two are so often confused when reading aloud. “The sum of $x$ and $5$, times three” is a product. “Three times $x$, plus five” is a sum.

Why naming the parts is worth doing

A question that says “the coefficient of $x$” points at exactly one number. A question that says “the 4” could mean anything on the page.

The vocabulary is what makes instructions short and unambiguous:

• Collect the like terms — you need to know what a term is.
• Factor out the common factor — you need to know what the factors are.
• The constant term is the $y$-intercept — you need to know which term that is.

It also organises the work. Splitting $7x - 4 + 2x + 9$ into its four terms before doing anything is what makes it clear that $7x$ and $2x$ can be combined while $-4$ and $9$ combine separately. The structure has to be visible before it can be used, and naming the parts is how it becomes visible.

Worked examples

Common mistakes

Practice problems