The Definition, Branches and Identities of Algebra

Algebra is one of the most important concepts in mathematical knowledge. There are several uses for algebraic identity in every area of a student’s life. A formula that holds true regardless of the value assigned to the variables in the equation is known as an algebraic identity. It means that the left-hand side (LHS) and right-hand sides (RHS) of the equation are always equal (RHS).

Let’s understand the definition of algebra.

Algebraic Identities: Definition

Algebraic identities are equations in algebra that are true regardless of the values of any variables. Algebraic equations consist of numbers, variables (unknown values), and mathematical operators (addition, subtraction, multiplication, division, etc.)

Algebra, geometry, and trigonometry are just a few mathematical disciplines that use algebraic identities. The primary purpose of these is to identify the polynomial factors. The effectiveness of solving issues can be increased by having a greater understanding of algebraic identities. One of the most vital uses of algebraic identities is the factorisation of polynomials.

Branches of Algebra

There are five different branches of algebra; let’s discuss them in detail.

1. Elementary Algebra – This branch of mathematics focuses on the basic characteristics of numbers, variables, constants, and their relationships.

2. Intermediate Algebra – This is algebra at the intermediate level. There are many more equations to answer in this algebra than in pre-algebra or elementary algebra. You will gain a better understanding of algebra by studying advanced algebra, which includes:

· Matrices

· Series

· Conic Sections

· Polynomial Situations

· Trigonometry

· Equalities

· Inequalities

· Sequences

· Probability

· Graphic Representation

· Rational Expression

3. Linear Algebra – Linear algebra is a branch of mathematics that has both applied and pure applications. It is concerned with vector space linear mappings. It also includes the study of planes and lines. It is the study of transformational properties in linear equation sets. In mathematics, it is almost universally used. It concerns linear equations, their representation in vector spaces, and linear function matrices.

4. Cumulative Algebra – It is a branch of mathematics that studies commutative rings and their ideals. Algebraic number theory and algebraic geometry both require commutative algebra. Differential order theory, topology, invariant theory, and generic topology are just a few domains where commutative algebra is applied. It has considerable significance in modern pure mathematics.

5. Abstract Algebra – One of the divisions of algebra is abstract algebra, which aims to find truths about algebraic systems independent of the nature of specific operations. These operations have unique characteristics in some cases. As a result, we can draw some inferences about the consequences of such characteristics. 

Algebra identities under the binomial theorem

The binomial theorem establishes a standard method for multiplying binomials or other words. 

The following is the general form of such basic algebra identities:

1. Identity-I: (a + b)² = a² + 2ab + b²

2. Identity-II: (a – b)² = a² – 2ab + b²

3. Identity-III: (a + b)³ = a³ + 3a²b + 3ab² + b³

4. Identity-IV: (a – b)³ = a³ – 3a²b + 3ab² – b³

5. Identity-V: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

6. Identity-VI: (x + a)(x + b) = x² + (a + b) x + ab

7. Identity-VII: (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

8. Identity-VIIII: (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴

Trinomial algebra identities

Equalities of trinomial basic algebra identities correspond to the corresponding equalities. By factoring and manipulating the terms (provided below), you can deduce such identities:

1. Identity-I: (a + b) (a + c) (b + c) = (a + b + c) (ab + ac + bc) – abc

2. Identity-II: a² + b² + c² = (a + b + c)² – 2(ab + ac + bc)

3. Identity-III: a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – ac – bc)

4. Identity-IV: (a – b) (a – c) = a² – (b + c)a + bc

Conclusion

These algebraic identities are primarily used to find polynomial factors. A deeper understanding of algebraic identities helps to improve the efficiency with which problem sums are solved. As a result, we can observe that an algebra identity has a formula that can fulfil any variable value. The basic algebraic identities are primarily helpful in solving a wide range of arithmetic issues.