An **algebraic expression** is an **expression** is made up of integer constants, variables, and **algebraic** operations (addition, subtraction, multiplication, division, and exponent. For example, 8x^{2} + 4xy – z is an **algebraic expression**.

We are now leading you from basic arithmetic to algebra.

In algebra, letters are used to represent numbers.

**Types of Algebraic expressions:**

There are 3 main branches of algebraic expressions which include:

- Monomial Expression
- Binomial Expression
- Polynomial Expression

**Monomial Expression:**

An algebraic expression which has only one term is known as a monomial.

Examples of monomial expression include 2×3, 4xy, etc.

**Binomial Expression:**

A binomial expression is an algebraic expression which has two terms.

**Example 1:**

Since 8x and 2x are like terms then,

8x+2x =10x

Since 8 and 2 are the multiples of x, therefore 8 and 2 are the coefficients.

**Example:2**

Add 8x and 2

Here both the terms are unlike so, the sum will be, 8x+2

**Polynomial Expression:**

Polynomial expression having more than one terms with non-negative integral exponents of a variable. Example 2x^{2}+5x+9-4x^{2}+2x -6

**Further branches of Expression:**

Including monomial, binomial and polynomial branches of expressions, an algebraic expression can also be categorized into two additional branches which are:

- Numeric Expression
- Variable Expression

**Numeric Expression:**

In numeric expression, never include any variable. These only consist of numbers and operations. Some of the examples of numeric expressions are 20+5, 15÷3, etc.

**Variable Expression:**

A variable expression is an expression which occupies variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x+y, 6ab + 25, etc.

An algebraic expression is a mathematical phrase that depends on numbers and/or variables. Though it cannot be solved because it does not contain an equals sign (=), it can be simplified. Algebraic equations, however, solve, which contain algebraic expressions separated by an equals sign.

**Following are the fundamental steps to simplify an algebraic expression:**

You need to simplify an algebraic expression before you evaluate it. This will make all your calculations much easier. Here are the main steps to follow for simplifying an algebraic expression:

- Remove parentheses by multiplying factors
- In terms with exponents. Use exponent rules to remove parentheses
- Merge like terms by adding coefficients
- Merge the constants.

Let’s work through an example.

In this expression, we can use the distributive property to get rid of the sets of parentheses. When simplifying an expression, the first thing you need to look for clear parentheses. We multiply the factors to the terms inside the parentheses.

**= 3(3+x)+4(2x+2)+(x ^{2})^{2}**

We have to get rid of the parentheses in the term with the exponents by using the exponent rules. When a term with an exponent is raised to a power, we multiply the exponents, so (x^{2})^{2} becomes x^{4}.

**= 9+3x+8x+8+x ^{2}**

The next step in simplifying we have to simplify the like terms and combine them. Here 3x and 8x are like terms, because they have the same variable with common powers, the first power since the exponent is understood to be 1. We can combine these two terms to get 11x.

**= 9+11x+8+x ^{4}**

**= 17+11x+x ^{4}**

Finally, our expression is simplified. Keep in mind one more thing that algebraic expression is usually written in a certain order. We start with the terms that have the largest exponents to the constants. Using the commutative property of addition, we rearrange the terms and put this expression in correct order, like this.

**= x ^{4}+11x+17**

**1. Difference between an algebraic expression and an algebraic equation:**

An algebraic equation can be solved and does include a series of algebraic expressions separated by an equals sign.

An algebraic expression is a mathematical phrase that can contain numbers and/or variables. It does not contain an equals sign and cannot be solved.

Expression:

3X + 9 or 4x-8

Equation:

4x +2=18 or 9x-2=14

**2. Solve an algebraic equation with exponents.**

If the equation has exponents, then you have to isolate the exponent on one side of the equation and then to solve by “removing” the exponent by finding the root of both the exponent and the constant on the other side. Let’s know how you do it:

- 2x
^{2}+ 6 = 14

First, subtract 12 from both sides.

- 2x
^{2}+ 6 -6 = 14 -6 = 0 - 2x
^{2}/2 = 8/2 = 0 - x
^{2}= 4

Solved by taking the square root of both sides, since that will turn x^{2} into x.

√x^{2} = √4

State both answers:x = 2, -2

**Solve an Algebraic Expression with subtraction**

Subtract (3x+2y) from (6x+7y)

= (6x+7y) – (3x+2y)

Now, we multiply negative sign to (3x + 2y):

= 6x+7y-3x-2y

Now, we arrange the expression as:

= 6x-3x+7y-2y

So, we got the answer 3x+5y

Example: 2

Subtract (4X^{2} – 2X + 6) from (2x^{2} + 5x + 9)

= (2x^{2} + 5x + 9) – (4X^{2} – 2X + 6)

= 2x^{2} + 5x + 9 – 4X^{2} + 2X – 6

Now, arrange the expression:

= 2x^{2} – 4X^{2} + 5x + 2X + 9 – 6

= -2x^{2}+7x+3

**Solve an Algebraic Expression with Multiplication**:

Example # 1

We have the algebraic expression (2x^{2})(3)

We multiply (2x^{2}) by 3

So we got the answer 6×2

Example # 2

Multiply (x + 2) by (3x – 5)

= x (3x – 5) + 2 (3x – 5)

= 3x^{2} – 5x + 6x – 10

= 3x^{2} + x – 10

**Solve an Algebraic expression with division:**

**Example # 1**

= -25×3 by 5x

= -25x^{3} / 5x= -5x

Example # 2

Divide (12x<sup>2</sup> + 3x + 9) by 3

=12x^{2}+3x +9 / 3

=12x^{2} + 3x^{3} + 9 / 3

=4x^{2} + x + 3

**Solve an algebraic expression with fractions.**

If you want to solve an algebraic expression with fractions, then you have to cross multiply the fractions, combining like terms, and then isolate the variable. Here’s an example of how you would do it:

- (x + 3)/6 = 2/3

First, cross multiply to remove the fraction. You have to multiply the numerator of one fraction by the denominator of the other.

- (x + 3) x 3 = 2 x 6 =
- 3x + 9 = 12

Now, combine like terms. Combine the constant terms, 9 and 12, by subtracting 9 from both sides.

- 3x + 9 – 9 = 12 – 9
- 3x = 3

Isolate the variable, x, by dividing both sides by 3 and you’ve got your answer.

- 3x / 3 = 3 / 3
- x =1

**Solve an algebraic expression that contains an absolute value.**

The absolute value is always positive.

For example, the absolute value of -7 (also known as |7|), is simply 7. To find the absolute value, you have to separate the absolute value and then solve for x twice, solving both for x with simply removed the absolute value, and for x when the terms on the other side of the equals sign have changed their signs from positive to negative and vice versa. Here’s how to do it.

- |6x +2| – 4 = 8 =0
- |6x +2| = 8 + 4 =0
- |6x +2| = 12 =0
- 6x + 2 = 12 =0
- 6x = 12
- x = 2

**Now, solve again by flipping the sign of the term on the other side of the equation after you’ve isolated the absolute value:**

- |6x +2| = 12 =0
- 6x + 2 = -12
- 6x = -12 -2
- 6x = -14
- 6x/2 = -14/2 =0
- x = -7

Now, just state both answers: x = 3, -7