Algebraic Expressions

 

Algebraic Expression

An algebraic expression (or) a variable expression is a combination of terms by the operations such as addition, subtraction, multiplication, division, etc. For example, let us have a look at the expression 5x + 7. Thus, we can say that 5x + 7 is an example of an algebraic expression. Here are more examples:

·         5x + 4y + 10

·         2x²y - 3xy²

·         (-a + 4b)² + 6ab

Variables, Constants, Terms and Coefficients

There are different components of an algebraic expression. Let us have a look at the image given below in order to understand the concept of Variables, Constants, Terms and Coefficients of any algebraic expression.

                                                                

In mathematics,

·         a symbol that doesn't have a fixed value is called a variable. It can take any value. In the above example that involved matchsticks, n is a variable and, in this case, it can take the values 1,2,3, ... Some examples of variables in Math are a, b, x, y, z, m, etc.

·         On the other hand, a symbol that has a fixed numerical value is called a constant. All numbers are constants. Some examples of constants are 3, 6, -(1/2), √5, etc.

·         A term is a variable alone (or) a constant alone (or) it can be a combination of variables and constants by the operation of multiplication or division. Some examples of terms are 3x², -(2y/3), √(5x), etc.

·         Here, the numbers that are multiplying the variables are 3, -2/3, and 5. These numbers are called coefficients.

 

                

Simplifying Algebraic Expressions

To simplify an algebraic expression, we just combine the like terms. Hence, the like variables will be combined together. Now, out of the like variables, the same powers will be combined together. For example, let us take an algebraic expression and try to reduce it to its lowest form in order to understand the concept better. Let our expression be:

x³ + 3x² − 2x³ + 2x − x² + 3 − x

= (x³ − 2x³) + (3x² − x²) + (2x − x) + 3

= −x³ + 2x² + x + 3

Hence, the algebraic expression x³ + 3x² − 2x³ + 2x − x² + 3 − x simplifies to −x³ + 2x² + x + 3.

 

Adding Algebraic Expressions

Here are some examples for adding algebraic expressions:

·         (x² + 2x + 3) + (2x² - 3x) = (x² + 2x²) + (2x + (-3x)) + 3 = 3x² - x + 3

·         (1.5ab + 3) + (2.5ab - 2) = (1.5ab + 2.5ab) + (3 + (-2)) = 4ab + 1

 

Subtracting Algebraic Expressions

To subtract two algebraic expressions, we add the additive inverse of the second expression to the first expression. Here are some examples for subtracting algebraic expressions:

·         (3x² - 5x) - (x² - 2x + 2) = (3x² - 5x) + (-x² + 2x - 2) = (3x² - x²) + (-5x + 2x) - 2 = 2x² - 3x - 2

·         (3ab + 4) - (2ab - 4) = (3ab + 4) + (-2ab + 4) = (3ab - 2ab) + (4 + 4) = ab + 8

 

Multiplying Algebraic Expressions

To multiply two algebraic expressions, we multiply every term of the first expression with every term of the second expression and combine all the products. Here are some examples of multiplying algebraic expressions.

·         ab (2ab + 3) = 2a²b² + 3ab

·         (x + 1) (x + 2) = x² + x + 2x + 2 = x² + 3x + 2

 

Dividing Algebraic Expressions

To divide two algebraic expressions, we factor the numerator and the denominator, cancel the possible terms, and simplify the rest. Here are some examples of dividing algebraic expressions.

·         2x² / (2x² + 4x) = (2x²) / [2x (x + 2)] = x / (x + 2)

·         (x² + 5x + 4) / (x + 1) = [ (x + 4) (x + 1)] / (x + 1) = x + 4

 

Algebraic Expressions Formula

Algebraic formulas are the derived short formulas that help us in solving the equations easily. They are just a rearrangement of the given terms in order to create a better expression that is easy to memorize. Find below a list of some of the basic formulas that are being used widely. Have a look at this page in order to understand the algebraic formulas better.

·         (a + b)² = a² + 2ab + b²

·         (a - b)² = a² - 2ab + b²

·         (a + b) (a - b) = a² - b²

·         (x + a) (x + b) = x² + x (a + b) + ab

·         (a + b)³ = a³ + 3a²b + 3ab² + b³

·         (a - b)³ = a³ - 3a²b + 3ab² - b³

·         a³ + b³ = (a + b) (a² - ab + b²)