How Many Algebraic Identities are There

Introduction

Algebra is an important chapter of basic maths. The algebraic identities are the equations which are valid of all the values of variables, and the equation will hold true irrespective of the value of the variables. Algebraic identities and expressions are simple mathematical equations that consist of numbers, variables which are unknown values, and mathematical operators such as addition, subtraction, multiplication, division etc. These equations are mainly used to find the factors of polynomials.

Two Variable Identities

The following identities in algebra have two variables. These identities can be verified very easily by expanding the square or the cube and performing polynomial multiplication.Here is an example to verify the first identity that is,

(a+b) ^{2}=(a+b) (a+b) =a^{2}+ab+ab+b^{2}=a^{2}+2ab+b^{2}

In the same way, we can also verify all the other identities.

I: (a+b)^{2}=a^{2}+2ab+b^{2}

II: (a-b)^{2}=a^{2}-2ab+b^{2

III: (a+b)(a-b) = a^{2}-b^{2}

IV: (a+b) ^{3}=a^{3}+b^{3}+3ab(a+b)

V: (a-b) ^{3}=a^{3}-b^{3}-3ab(a-b)

Three Variable Identities

The algebra identities for three variables also have derived the way as the two variable identities were derived. Moreover, these identities are helpful to easily work across the algebraic expressions with very less number of steps.

I:(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca

II:a^{2}+b^{2}+c^{2}=(a+b+c)^{2}-2(ab+bc+ac)

III:a^{3}+b^{3}+c^{3}=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ca)

IV:(a+b)(b+c)(c+a)=(a+b+c)(ab+ac+bc)-2abc

V: (x+a)(x+b)=x^{2}+x(a+b)+ab

Factorization Identities

Algebraic identities are helpful in factorizing an algebraic expression effortlessly. Using these identities, higher algebraic expressions can be easily factorized using the basic algebraic identities. The list below presents you with a set of algebraic identities that are helpful for the factorization of any polynomials.

I: a^{2}-b^{2}=(a-b) (a+b)

II: x^{2}+x(a+b)+ab=(x+a)(x+b)

III: a^{3}-b^{3}=(a-b)(a^{2}-ab+b^{2})

IV: a^{3}+b^{3}=(a+b) (a^{2}-ab+b^{2})

Difference Between an Algebraic Expression and Identities

Algebraic expression refers to any expression that has variables and constants. The value of a variable can be anything in an expression. Therefore, the expression value might change if we use different values of the variable.

On the contrary, algebraic identity is true for all the values of the variables. Every equation that you come across is not an identity, but every algebraic identity is an equation.

Verifying Algebraic Identity

These identities are usually verified using the substitution method. In this method, we used to substitute the values of the variables and then perform the arithmetic operation.

Another method that can be used to verify an algebraic identity is the activity method. In this method, one necessarily requires knowledge of Geometry and some materials that are needed to prove the identity.

Binomial Theorem

As the power of the polynomial increases, the expansion of it becomes lengthy and boring to calculate. This theorem is defined as a standard way of expanding a binomial expression raised to a large power or other terms which can be annoying. A polynomial equation with just two terms, mostly having a plus or a minus sign in between is a Binomial expression. It is used in algebra, probability etc. It is possible to expand any non-negative power of the binomial expression (x + y) into a sum of the form,

(x+y)^{n}=^{n}C_{0}x^{n}y^{0}+^{n}C_{1}x^{n-1}y^{1}+......... ^{n}C_{n-1}x^{1}y^{n-1}+^{n}C_{n}x^{0}y^{n}

where, n ≥ 0 is an integer and each is a positive integer which is known as binomial coefficient. When the exponent is 0 then the power expression correspondingly to it is 1.

Conclusion

Algebraic identities are used in various aspects of mathematics, such as algebra, geometry, trigonometry etc. There are applications of algebraic identities in advanced mathematical equations, analysis and even research-based concepts. You need to have great understanding to efficiently solve these sums. One of the most important applications of algebraic identities is in the factorization of polynomials. If you're a mathematics student, you need to be proficient with these equations, as they are very helpful in solving engineering and scientific related problems.