Algebraic expression is a mathematical phrase which contain numbers, operators, and variables to represent operations. It is made up of terms and terms are each separate values in an expression which can be separated by mathematical operations (+, , *, /). An algebraic expression is a mathematical sentence where there is no equal sign. For example, 7 + 9, 5xyz, 5x$^{2}$ + 7xyz + 36z  156 etc., In this expression 5x$^{2}$ + 7xyz + 36z  156, there are four terms 5x$^{2}$, 7xyz, 36z and 156. Terms may consist of variables and coefficients, or constants.

In an expression there are three parts, the variable, constant, and coefficient.
In algebraic expressions letters represent variables. In variable we can substitute one or more numbers for the letters in the expression. It is an unknown or changing number. Usually 'x' is used in an algebraic expression.
Constants contain only numbers in an algebraic expression. They are the terns without variables. We call them constants as their value never changes in an algebraic expression.
For example: 15x + 7y + 2 : Constant term is '2'.
35x$^{2}$ + 16y$^{2}$  9 : Constant term is 9
The number that is multiplied by the variable in an algebraic expression is known as Coefficient.
In 5x$^{2}$ + 7y  4z + 5xy + 10, the coefficient of the first term is 5, the coefficient of the second term is 7, the coefficient of third and fourth term is 4 and 5 respectively.
A monomial is an expression in algebra that contains only one term and includes numbers and variables that are multiplied together.
Example: 5xy, 12a, z. A monomial cannot have a fractional or negative exponent. A monomial multiplied by a monomial (constant) is a monomial.
Binomial is an algebraic expression (or a polynomial) containing two terms that are not like terms. Examples: 5x$^{3}$ + 2x, x  25, 7x + 5.
A trinomial is a polynomial with three terms.
Examples of trinomials are 15x$^{3}$ + 3x$^{2}$ + 31, 5x$^{7}$  18x$^{4}$ + 55x.
Polynomials are algebraic expressions that include real numbers and variables. Variables include addition, subtraction and multiplication. Division and square roots cannot be involved in the variables. Polynomials contain more than one term.
Example: 8y, 9y  5y$^{2}$, 3x$^{2}$ + 5x + 9 etc.,
The degree of the term is the exponent of the variable: 7x$^{2}$ has a degree of 2.
When the variable does not have an exponent  always understand that there's a '1' .
A rational expression is a fraction in which the numerator and\or the denominator are polynomials. An algebraic expression, that is a quotient of two other algebraic expressions, is called a rational algebraic expression. If P and Q are algebraic expressions and Q $\neq$ 0 then the expression $\frac{P}{Q}$ is called a rational algebraic expression.Examples: $\frac{6}{x1}$, $\frac{Z^{2} 5}{Z^{2} + 4}$.
When a numerical expression involves two or more operations, there is a specific order in which these operations must be performed. Here we will learn to add, subtract, multiply and divide algebraic expression.
Addition and Subtraction
Addition and Subtraction
Group like terms. Only like terms can be added.
Add or subtract the coefficients of the grouped like terms.
Example: Add 5x  2 and 5 (2x  4)
5x  2 + 5 (2x  4)
= 5x  2 + 10x  20
= (5x + 10x)  (2 + 20)
= 15x  22
Multiplication
When multiplying algebraic expressions, follow the steps below:
Step 1: List the coefficients and variables of each term separately.
Step 2: Multiply the coefficients and constants.
Step 3: Multiply the variables.Example: Solve : 2(x + 5) (2x  5)
2 (x + 5) (2x  5) = (2x + 10) (2x  5)
= 2x (2x  5) + 10 (2x  5)
= 4x$^{2}$  10x + 20x  50
= 4x$^{2}$ + 10x  50
$\Rightarrow$ 2 (x + 5) (2x  5) = 4x$^{2}$ + 10x  50
Division
When dividing algebraic expressions, follow the steps below:
Step 1: List the coefficients and variables of each term separately.
Step 2: Divide the product of the coefficients and constants in the numerator, by the product of the coefficients and constants in the denominator.
Step 3: Cancel the like variables in the numerator, by those in the denominator.Example: $\frac{x^{2}9}{x^{2}+3x}$
= $\frac{x^{2}3^{2}}{x(\frac{x^{2}}{x}+\frac{3x}{x})}$
= $\frac{(x+3)(x3)}{x(x+3)}$
= $\frac{x3}{x}$
5x  2 + 5 (2x  4)
= 5x  2 + 10x  20
= (5x + 10x)  (2 + 20)
= 15x  22
Multiplication
When multiplying algebraic expressions, follow the steps below:
Step 1: List the coefficients and variables of each term separately.
Step 2: Multiply the coefficients and constants.
Step 3: Multiply the variables.Example: Solve : 2(x + 5) (2x  5)
2 (x + 5) (2x  5) = (2x + 10) (2x  5)
= 2x (2x  5) + 10 (2x  5)
= 4x$^{2}$  10x + 20x  50
= 4x$^{2}$ + 10x  50
$\Rightarrow$ 2 (x + 5) (2x  5) = 4x$^{2}$ + 10x  50
Division
When dividing algebraic expressions, follow the steps below:
Step 1: List the coefficients and variables of each term separately.
Step 2: Divide the product of the coefficients and constants in the numerator, by the product of the coefficients and constants in the denominator.
Step 3: Cancel the like variables in the numerator, by those in the denominator.Example: $\frac{x^{2}9}{x^{2}+3x}$
= $\frac{x^{2}3^{2}}{x(\frac{x^{2}}{x}+\frac{3x}{x})}$
= $\frac{(x+3)(x3)}{x(x+3)}$
= $\frac{x3}{x}$
Solved Examples
Question 1: Write each phrase as an algebraic expression for the following.
The sum of seven and a number.
Twelve decreased by a number.
Five less than a number x.
Six more than twice a number.
Product of eight and a number.
Solution:
The sum of seven and a number.
Twelve decreased by a number.
Five less than a number x.
Six more than twice a number.
Product of eight and a number.
Solution:
Statement 
Algebraic Expression 
The sum of seven and a number  7 + x 
Twelve decreased by a number  12  x 
Five less than a number x  5  x 
Six more than twice a number  2x + 6 
Product of eight and a number  8x 
Question 2: Simplify: 12x + 3 + x  1  5x
Solution:
Arrange the terms:
3  1 + 12x + x  5x
Combining like terms:
2 + 8x
$\Rightarrow$ 12x + 3 + x  1  5x = 2 + 8x
Solution:
Arrange the terms:
3  1 + 12x + x  5x
Combining like terms:
2 + 8x
$\Rightarrow$ 12x + 3 + x  1  5x = 2 + 8x