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Parts of an Expression

Algebraic expressions are combinations of variables , numbers, and at least one arithmetic operation.

For example, $2 x + 4 y - 9$ is an algebraic expression.

Math diagram

Term: Each expression is made up of terms. A term can be a signed number, a variable, or a constant multiplied by a variable or variables.

Factor: Something which is multiplied by something else. A factor can be a number, variable, term, or a longer expression. For example, the expression $7 x (y + 3)$ has three factors: $7$ , $x$ , and $(y + 3)$ .

Coefficient: The numerical factor of a multiplication expression that contains a variable. Consider the expression in the figure above, $2 x + 4 y - 9$ . In the first term, $2 x$ , the coefficient is $2$ : in the second term, $4 y$ , the coefficient is $4$ .

Constant: A number that cannot change its value. In the expression $2 x + 4 y - 9$ , the term $9$ is a constant.

Like Terms: Terms that contain the same variables such as $2 m$ , $6 m$ or $3 x y$ and $7 x y$ . If an expression has more than one constant terms, those are also like terms.

Expression	Word Phrases
$n + 5$	Sum of a number and $5$
$m - 7$	Difference of a number and $7$
$6 x$	Product of $6$ and a number
$y \div 9$	Quotient of a number and $9$

Example:

Identify the terms, like terms, coefficients, and constants in the expression.

$9 m - 5 n + 2 + m - 7$

First, we can rewrite the subtractions as additions.

$9 m - 5 n + 2 + m - 7 = 9 m + (- 5 n) + 2 + m + (- 7)$

So, the terms are $9 m$ , $(- 5 n)$ , $m$ , $2$ , and $(- 7)$ .

Like terms are terms that contain the same variables.

$9 m$ and $9 m$ are a pair of like terms . The constant terms $2$ and $- 7$ are also like terms.

Coefficients are the numerical parts of a term that contains a variable.

So, here the coefficients are $9$ , $(- 5)$ , and $1$ . ( $1$ is the coefficient of the term $m$ .)

The constant terms are the terms with no variables, in this case $2$ and $- 7$ .

Algebraic expressions must be written and interpreted carefully. The algebraic expression $5 (x + 9)$ is not equivalent to the algebraic expression, $5 x + 9$ .

See the difference between the two expressions in the table below.

Word Phrases	Algebraic Expression
Five times the sum of a number and nine	$5 (x + 9)$
Nine more than five times a number	$5 x + 9$

In writing expressions for unknown quantities, we often use standard formulas. For example, the algebraic expression for "the distance if the rate is $50$ miles per hour and the time is $T$ hours" is $D = 50 T$ (using the formula $D = R T$ ).

An expression like $x^{n}$ is called a power. Here $x$ is the base, and $n$ is the exponent. The exponent is the number of times the base is used as a factor. The word phrase for this expression is " $x$ to the $n^{th}$ power."

Here are some of the examples of using exponents.

Word Phrases	Algebraic Expression
Seven times $m$ to the fourth power	$7 m^{4}$
The sum of $x$ squared and $12$ times of $y$	$x^{2} + 12 y$
$x$ cubed times $y$ to the sixth power	$x^{3} \cdot y^{6}$