Algebraic Expressions and Identities

Algebraic Expressions and Identities: In mathematics, An algebraic expression is a combination of variables and constants. The value of an algebraic expression can vary because the variables can take different values. On the other hand, an algebraic identity is an equation that holds true for all possible values of the variables involved.

To understand these terms, we need to have an understanding of terms, factors, and coefficients. In this article, we have discussed algebraic expressions and identities with their examples in detail.

Algebraic Expressions are combinations of variables, constants, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). They are used to represent mathematical relationships and can be simplified or evaluated for specific values of the variables. An algebraic expression does not have an equality sign, differentiating it from an equation.

Terms: In algebra, a term can be a variable a constant, or a constant multiplied by a variable.

Example: 3x, 4, xy.

Factors: In algebra, factors are all the possible parts of the product.

Note: 1 is a factor for everything.
Example 1: Factors of 5x are 1,5,x, and 5x.
Example 2: Factors of 6x(y+7)  are 1,6,x, y+7.

Coefficients: In algebra, When a term is formed when a constant is multiplied by a variable or variables, that constant is called a coefficient.

Example1: 5x: In this term, 5 is the coefficient.
Reason: As 5 is a constant and is being multiplied to a variable ‘x’ by definition ‘5’ is called a coefficient. 

Example 2: 3x+4y: In this expression, 3, 4 are coefficient.
Reason: s 3 and 4 are constants and are being multiplied to a variable ‘x’ and ‘y’ so by definition ‘3,4’ are called as coefficients. 

An algebraic expression is an expression that is made up of variables and constants, along with algebraic operations (like subtraction, addition, multiplication, etc.). Expressions are made up of terms. 

Example: 5x+20y, 6-8x.

Expressions in Algebra are divided into three types based on the number of Terms involved in the expression. These types are:

• Monomial Expression
• Binomial Expression
• Polynomial Expression

1. Monomial Expression

Algebraic expressions that contain only one term are called Monomial Expressions.
Examples: 5x, 10y, 25yz, etc.

2. Binomial Expression

Algebraic expression which has two terms (different or unlike terms) is called binomial expression. 
Examples: 30xy+60, 25x+24y, 7+8yz, etc.

3. Polynomial Expression

Algebraic expression which contains more than one term with non-negative integer exponents is called Polynomial Expression.
Examples: 2x+3y+4z, 10x+20y+45,etc.

Algebraic Identities are equations that hold true for all values of the variables involved. They are often used to simplify expressions and solve equations. Identifying and applying these identities can make complex algebraic manipulations more manageable.

Example: Implement the first Identity on x = 4, and y = 3 

Solution: 

Applying the identity:
              L.H.S => (x+y)² = (4+3)²                           
                                      =  (7)²
                                                  = 49

             R.H.S => 4² +3²+ 2 (4) (3) = 49 

 As L.H.S = R.H.S, this identity is verified and true.

Example: Implement the second Identity on x = 4, and y = 3 

Solution: 

Applying the identity:
                 L.H.S => (x – y)² = (4 – 3)²                          
                                           =  (1)²
                                           =  1

               R.H.S => 4² + 3² – 2 (4) (3) = 1

As L.H.S = R.H.S, this identity is verified and true.

Example: Implement the third Identity on x = 4, and y = 3 

Solution: 

Applying the identity:
                 L.H.S => (x + y)(x – y) = (4 + 3)(4 – 3)                                                      
                                                    = (7)(1)  
                                                    = 7 

                 R.H.S :  x² – y²   = (4)^{2 –}  (3)²

                                                       = 16 – 9      

                                         = 7

 As L.H.S = R.H.S, this identity is verified and true.

Example: Implement the fourth Identity on x = 3, y = 4, and z = 5.

Solution: 

Applying the identity:

                  L.H.S => (x + y + z)² = (3 + 4 + 5)² 

                                                   = (12) ² 

                                                   = 144                          
                                                    

                  R.H.S => x2 – y2   = (3)² + (4)² + (5)² + 2(3)(4) + 2(4)(5) + 2(5)(3)
                                               = 144

 As L.H.S = R.H.S, this identity is verified and true.

Using the above Identities we can derive many identities, some of the popularly used identities are written below:

1. Cube of a Sum

[Tex](a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 [/Tex]

2. Cube of a Difference

[Tex](a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3 [/Tex]

3. Sum of Cubes

[Tex]a^3 + b^3 = (a + b)(a^2 – ab + b^2) [/Tex]

4. Difference of Cubes

[Tex]a^3 – b^3 = (a – b)(a^2 + ab + b^2) [/Tex]

1. Simplify (3x + 4)²

[Tex](3x + 4)^2 = (3x)^2 + 2 \cdot 3x \cdot 4 + 4^2 = 9x^2 + 24x + 16 [/Tex]

2. Factorize: x² – 16

[Tex]x^2 – 16 = (x + 4)(x – 4)[/Tex]

3. Expand (2a – 3b)³

[Tex](2a – 3b)^3 = (2a)^3 – 3 \cdot (2a)^2 \cdot 3b + 3 \cdot 2a \cdot (3b)^2 – (3b)^3 = 8a^3 – 36a^2b + 54ab^2 – 27b^3 [/Tex]

What are the key differences between algebraic expressions and identities?

Algebraic expressions consist of variables and constants, and their values can change based on the variables. In contrast, algebraic identities are equations that are universally true for all variable values.

How do you simplify algebraic expressions?

Simplifying algebraic expressions involves combining like terms, using the distributive property, and applying arithmetic operations to reduce the expression to its simplest form.

What are the most important algebraic identities to know?

Some of the most important algebraic identities include the square of a sum (a + b)², square of a difference (a – b)², and the product of a sum and difference (a + b)(a − b). Others include the cube of a sum (a + b)³, cube of a difference (a – b)³, and the sum and difference of cubes.

How can algebraic identities be used to factor polynomials?

Algebraic identities are useful tools for factoring polynomials. For example, the identity a² – b² = (a+b) (a-b) can be used to factor a difference of squares, while identities for the sum and difference of cubes can help factor cubic polynomials.

Why are algebraic identities important in mathematics?

Algebraic identities are important because they provide shortcuts for simplifying and solving complex algebraic problems. They are fundamental in algebra and are used extensively in higher mathematics, including calculus and linear algebra.