Parts of an Expression
Algebraic expressions are combinations of variables , numbers, and at least one arithmetic operation.
For example, $2x+4y9$ is an algebraic expression.
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 $7x\left(y+3\right)$ has three factors: $7$ , $x$ , and $\left(y+3\right)$ .
Coefficient: The numerical factor of a multiplication expression that contains a variable. Consider the expression in the figure above, $2x+4y9$ . In the first term, $2x$ , the coefficient is $2$ : in the second term, $4y$ , the coefficient is $4$ .
Constant: A number that cannot change its value. In the expression $2x+4y9$ , the term $9$ is a constant.
Like Terms: Terms that contain the same variables such as $2m$ , $6m$ or $3xy$ and $7xy$ . If an expression has more than one constant terms, those are also like terms.










Example:
Identify the terms, like terms, coefficients, and constants in the expression.
$9m5n+2+m7$
First, we can rewrite the subtractions as additions.
$9m5n+2+m7=9m+\left(5n\right)+2+m+\left(7\right)$
So, the terms are $9m$ , $\left(5n\right)$ , $m$ , $2$ , and $\left(7\right)$ .
Like terms are terms that contain the same variables.
$9m$ and $9m$ 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 $9$ , $\left(5\right)$ , 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 $5\left(x+9\right)$ is not equivalent to the algebraic expression, $5x+9$ .
See the difference between the two expressions in the table below.
Word Phrases  Algebraic Expression 
Five times the sum of a number and nine 
$5\left(x+9\right)$

Nine more than five times a number 
$5x+9$

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 $D=50T$ (using the formula $D=RT$ ).
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 ${n}^{\text{th}}$ 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 $y$ 
${x}^{2}+12y$

$x$ cubed times $y$ to the sixth power 
${x}^{3}\cdot {y}^{6}$
