Algebraic Expression

What is an Algebraic Expression?

In Algebra, we commonly see x, y and others. Why do we use letters? How can we solve it?

An algebraic expression is a combination of integer constants, variables, exponents and algebraic operations such as addition, subtraction, multiplication and division. 5x, x + y, x-3 and more are examples of algebraic expression. A constant is any set of numbers. A variable is a letter used to represent an unknown value. In solving an algebraic expression, simply combine like terms.

Algebraic Expression Operations

Algebraic Expression – Addition and Subtraction

In addition or subtraction of algebraic expression, it is important that the terms are like terms. Like terms are defined as the same variable and raised in the same power. Let’s take this example: x + 5 + 3x + 5y – 2 + 5.

x + 5 + 3x + 5y – 2 + 5             Group all the similar terms

x + 3x + 5y – 2 + 5                    Combine all like terms and same variables.

4x + 5y + 3

Let’s look at another: 5x + x^2 – 3x + 5

5x + x^2 – 3x + 5                     Group all similar terms

x^2 + 5x -3x + 5                      Combine like terms

x^2 + 2x + 5                             Observe that x^2 is not added to 2x. x^2 is in second degree while 2x is not. They are same variable but different degree.

Algebraic Expression – Multiplication

Multiplying an algebraic expression involves distributive property and index law. Let’s use this example: 5 multiplied to x is 5x. In multiplying, having a like term is not applied.

Let’s see another example: x(x+1)

x(x+1)                                      Expand the following using the distributive law

x(x) + x(1)

x^2 + x

Another example: (x-2)^2

(x-2)^2                                     This equation means that (x – 2) is multiplied with (x – 2)

(x – 2)(x – 2)                             Use distributive law

x(x) +x(-2) – 2(x) – 2(-2)

x^2 – 2x – 2x + 4

x^2 – 4x + 4

Note: In multiplication, do not forget to follow distributive law.

Algebraic Expression – Division

Dividing an algebraic expression is simplifying the term.

Look at this example: 30a^3b^2 divided by 5a^2b^3

30a^3b^2                                 To understand more about this term, write it this way

5a^2b^3

30 aaa  bb  
5  aa  bbb                                  Divide the constant then cancel the variables from top to bottom or subtract the exponent from top to bottom. All positive exponents are above and the negatives will be placed below.

 6a
  b

Another example:

(x + 1) ( x + 2)                         Simplify the algebraic expression by cancelling
(x + 2) (x – 3)

(x + 1)
(x – 3)

In solving equations having algebraic expression like this example 3/x + 4/(x + 1), the equations needs to have the same denominator. For this equation, multiply both equations by x(x + 1), then cancel.

3 (x) (x + 1)   =  3 (x+1)
X

4 (x) (x + 1)    = 4x                  After cancelling both equations
(x+1)

3 (x+1) + 4x

3x + 3 + 4x

7x + 3