Algebra is an important branch of mathematics that involves numbers, equations, and mathematical analysis. There are different methods and formulas for solving algebraic problems. It is one of the most important chapters of mathematics. Without a proper understanding of algebra, solving other questions is challenging. 

For better understanding, algebra is divided into five branches:

• Elementary algebra: It covers basic algebra topics like variables, algebraic problems, equations, linear equations, etc. 

• Advanced algebra: Also known as intermediate algebra, this includes higher levels of equations, like inequality equations, linear graphing equations, trigonometry, quadratic equations, matrices, etc. 

• Abstract algebra: This branch of algebra analyses algebraic equations independently according to the specific nature of algebraic operations. It covers topics like sets, inverse elements, identity elements, etc.

• Linear algebra: This involves studying the lines, linear equations, and planes. It covers topics like linear equations, computations, relations, vector spaces, etc.
• Commutative algebra: It studies the commutative rings of algebraic equations. It forms the basis for important algebraic theories, like algebraic number theory, invariant theory, order theory, etc.

Expansion is a mathematical function of expanding the problem by removing the brackets from the equation. It becomes crucial to remove brackets, mainly to solve equations that need to be simplified to get the solution. 

Following are a few formulas and identities used in the expansion of algebraic equations:

• (x + y) (x + z) = x^2 + (y + z) x + yz
• (x + y) (x – z) = x^2 + (y – z) x – yz 
• (x – y) (x + z) = x^2 – (y – z) x – yz
• (x – y) (x – z) = x^2 – (y + z) x + yz
• (x + y)^2 = x^2 + 2xy + y^2
• (x – y)^2 = x^2 – 2xy + y^2
• (x + y) (x – y) = x^2 – y^2

Factorisation is the act of factoring a number or a polynomial into various numbers or factors, such that when these factors are multiplied, they give the original number or polynomial. Factorisation is the opposite of expansion. 

Following are the various methods of factorisation:

• Grouping method: It means arranging the even-numbered polynomials in groups, such that each group has a common factor. 
• Common factors method: It means taking out the common factors and then dividing the polynomial. 
• Splitting the middle terms method: This method involves splitting the middle term of the trinomial polynomial arranged in descending order, such that their product is equal to the product of the first and last terms.
• Difference of two squares method: It involves using the identity x^2 – y^2 = (x + y) (x – y) to solve the polynomial.
• Difference or sum of two cubes method: It involves using the identity of cubes to solve the polynomial equations.

A group of linear equations in the same two variables is called simultaneous linear equations in two variables. The same pair of two variables solve these linear equations. Simultaneous linear equations can either have one solution, infinite solutions, or no solution. 

Following are the methods for solving simultaneous linear equations in two variables:

• • Substitution method: In this method, find the value of any one variable in the form of another variable for the first linear equation. Then substitute the value of the variable in the second equation to find the numerical value of the two variables. 
  • Elimination method: Multiply both the equations with a number such that one of the variables of both the equations becomes equal. Then subtract or add the equations to eliminate that variable and find the value of the two variables.
  • Cross product method: This method involves converting the linear equations into a general form to apply the cross product method to find the value of the two variables.

Exponent represents the power of the polynomial. For example, x^y represents that polynomial x is multiplied by itself y number of times, where y is the exponent and x is the polynomial or base.

There are 3 laws of exponents:

• Product law: This law states that the solution to polynomial ×^y × x^z is x^(y+z)
• Quotient law: This law states that the solution to polynomial x^y ÷ x^z is x^(y-z)
• Power law: According to this law, the solution to polynomials (x^y)^z is x^yz.

Algebra is an important chapter for class 9th mathematics and an essential topic for advanced mathematics. Sound knowledge of this chapter will help you learn other chapters like geometry, equations, differentiation, etc.

1. What do you mean by Algebra?
   Algebra is a branch of mathematics that involves studying numbers, equations, geometry, mathematical analysis, etc.
2. What are the expansion identities for cubic equations?
   (x + y)^3 = x^3 + 3xy(x + y) + y^3
   (x – y)^3 = x^3 – 3xy(x – y) – y^3
3. What are formulas used in the factorisation methods of difference or sum in two cubes?
   It involves using the identities of cubes to solve the polynomial equations, x^3 + 3xy(x + y) + y^3 = (x + y)^3 and x^3 – 3xy(x – y) – y^3 = (x – y)^3.
4. Which method is the best to solve simultaneous linear equations?
   Substitution and elimination methods are the easiest and most used methods to solve simultaneous linear equations.