How to Verify Algebraic Identities

Algebraic Identities: One of the most significant aspects of Elementary Mathematics is algebra. It is introduced to students in the lower years of secondary education and continues through senior secondary and into higher education. Algebraic Identities is a broad topic with applications in almost every aspect of a student’s life. An algebraic identity is a formula that holds true no matter what value is applied to the variables in the equation. It signifies that the equation’s left-hand side (LHS) is always equal to the right-hand side (RHS).

Definition of Algebraic Identities:

Algebraic identities are equations that hold true for all possible values of variables. Numbers, variables (unknown values), and mathematical operators make up algebraic identities and expressions (addition, subtraction, multiplication, division, etc.) 

Algebraic identities are employed in many areas of mathematics, including, A=πr² algebra, geometry, trigonometry etc. These are mostly used to calculate polynomial factors. A deeper understanding of algebraic identities helps to improve the efficiency with which problem sums are solved. The factorisation of polynomials is one of the most important applications of algebraic identities. 

What are Algebraic Identities?

An identity is a mathematical equation that holds true for all possible values of the variables. For all values of the variable, algebraic identities are equations in which the value of the left-hand side of an equation equals the value of the right-hand side of the equation. 

Example: Consider the linear equation ax+b=0. 

Here, the left-hand side and right-hand side of the above equations are the same when x= -b/a, Hence, it is not an identity, but it is an equation.

In (a + b)² = a² + 2ab + b²

we know that it is true for all values of variables a  and b. So, it is an identity.

Standard Identities in Algebra:

In several disciplines of mathematics, we have some standard identities to employ. The Binomial theorem is used to generate all of the standard identities. 

The following are four standard algebraic identities:

Algebraic identity of the square of the sum of two terms (identity-1)

(x + y)² = x² + 2xy + y²

Algebraic identity of a square of the difference of two terms (identity-2)

(x – y)² = x² – 2xy + y²

Identity-3: algebraic identity of difference of two squares

(x + y)(x – y) = x² – y²

Identity-4: algebraic identity (x+x)(x+y) 

(x + x)(x + y) = x² + x(x + y) + xy

Factorization Algebraic Identities :

Algebraic identities are extremely useful for factoring algebraic expressions quickly. Using the algebraic identities below, the given expression representing the expanded form of the formula can be translated and represented as a set of factors. Higher algebraic identities like  a⁴ – b⁴  can be easily deduced from the basic algebraic identities a² – b².

The list below contains a number of algebraic identities that can be used to factor polynomials. 

 x² – y² = (x – y)(x + y)

 x² + x(x + y) + xy = (x + x)(x + y)

 x³ – y³ = (x – y)(x² + xy + y²)

 x³ + y³ = (x + y)(x² – xy + y²)

 x⁴ – y⁴ = (x – y)(x + y)(x² + y²)

Three-variable Algebraic Identities:

The binomial expansion formula was also used to generate algebraic identities for three variables. Furthermore, these identities are useful in reducing the number of steps required to work across algebraic expressions.

x² + y² + z² = (x + y + z)² – 2(xy + yz + xz)

x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – zx – yz)

(x + y)(x + z)(y + z) = (x + y + z)(xy + xz + yz) – 3xyz

How can an Algebra identity be verified?

There are two simple approaches to verify algebraic identities. One method is to change the values of the variables in the algebraic identities. On either side of the equals to sign in an algebra identity, there is an expression. We can try to get the same answer on both sides by substituting the values on either side of the equals sign. Another way is to validate the algebra identity by manipulating and simplifying the left-hand side of the equation to produce the right-hand side.

Methods to Solve Identities:

We can check algebraic identities by substituting values into variable positions and attempting to make both sides equal. i.e LHS = RHS.

Example:

(x – 2) (x + 2) = x² – 2²

Now we have to put the values in place of a.

starting with x = 1, (-1) x (3) = -3

then we will put x = 2, 0 x 4 = 0

Here we got x = 1 and x = 2 as the value which satisfy the given question.

Another method is by manipulating identities which is  commonly used:

 (x + y)²=   x²  + y²  + 2xy

(a – b)² = a²+ b² – 2ab     

(a + b)(a – b) =a² – b²   

(x + a)(x + b) = x² + (a + b)x + ab

Proof of algebraic identities:

1. (x + y)² = (x + y) (x + y)

                = (x + y) (x) + (x + y) (y)

                = x² + xy + xy + y²

                = x² + 2xy + y²

Hence, LHS = RHS.

2. we can write :(x – y)² = (x – y) (x – y)

                = (x – y) (x) + (x – y) (y)

                = x² – xy – yx + y²

                = x² – 2xy + y²

Hence, LHS = RHS.

3 . (x + y) (x – y) = x (x – y) + y (x – y)

                            = x² – xy + xy – y²

                            = x² – y²

Hence, LHS = RHS.

Conclusion:-

Many mathematical calculations can benefit from algebra identities. Factorization, trigonometry, integration and differentiation, quadratic equations, and other topics may be covered.