Algebra is a branch of mathematics that involves studying numbers, equations, geometry, and mathematical analysis. 

For better understanding, algebra is divided into five branches:

1. Elementary algebra: It covers basic algebra topics like variables, solving algebraic problems and equations, solving linear equations, etc. 
2. Advanced algebra: Also known as intermediate algebra, this algebra has higher equations to solve like inequality equations, graphing linear equations, trigonometry, quadratic equations, matrices, etc. 
3. Abstract algebra: This branch of algebra analyses the algebraic equations independently to the specific nature of operations. It covers topics like sets, inverse elements, identity elements, etc.
4. Linear algebra: The branch of algebra that involves studying linear equations and planes is linear algebra. It covers topics like linear equations, computations, relations, vector spaces, etc. 
5. Commutative algebra: It studies the commutative rings of algebraic equations. It forms the basis for critical algebraic theories like algebraic number theory, invariant theory, order theory, etc.

A linear inequality is a mathematical statement with both sides not equal to each other. For example: y + 5 > 10. The solution set of linear inequalities is a subset of a universal set with solutions that satisfy the inequality. 

Following are properties of linear inequalities:

• Add or subtract a number from both sides of inequality results in an equivalent inequality. 
• Multiply or divide a positive number from both sides of inequality results in an equivalent inequality.
• Multiply or divide a negative integer from both sides of inequality results in inequality with a reversed sign like greater to sign changes to lesser signatures.

The factorisation is the act of factoring a number or a polynomial into various numbers of factors. When these factors are multiplied, they give the original number or polynomial. The factorisation is generally considered opposite to the properties of expansion.

Two theorems of factorisation of polynomials:

• Remainder theorem: If a polynomial is divided by a factor (x – a), then the remainder of the polynomial is f(x).
• Factor theorem: For a polynomial f(x) and number b, (x – b) is a factor of the polynomial if f(b) is equal to 0.

Quadratic equations are equations that can be written in the general form of ax^2 + b× + c. The maximum power of the quadratic equations is 2. For any quadratic equation, there are two roots, α and β, whose sum is equal to –b/a and multiplication is equal to c/a where the coefficient of x^2, b is the coefficient of x. The “c” is the constant term in the polynomial. 

Discriminant (D) of the polynomial is calculated as b^2 – 4ac. The following are the properties of discriminant:

• D = 0, roots are equal, and real numbers.
• D > 0, roots are distinct and real numbers. 
• D < 0, roots are imaginary in nature.
• D = perfect square, roots are distinct and rational numbers.

For any two numbers a and b, a/b is the ratio of a and b. When two or more ratios are multiplied by each other term-wise, then the ratio so obtained is known as the compound ratio. 

Proportion is a term used for four numbers such that the ratio of the first and second number is equal to the ratio of the third and fourth number. It is represented as a/b = c/d, where a and d terms are known as extreme terms while b and c terms are known as middle terms.

A rectangular arrangement of numbers in the form of columns and rows is known as the matrix. The matrix order is depicted in rows × columns like a matrix with three columns and four rows with the order 4×3. 

Two matrices are equal if the order is the same for both the matrices and the matrices’ elements are also the same.

A pair of points located on the plane are known as coordinates, and the geometrical formulas used on these coordinates are known as coordinate geometry.

Distance between two coordinates (a, b) and (c, d) is calculated using the following formula: 

Distance = √[c – a)^2 +(d – b)^2]

Arithmetic progression (AP) is a sequence of numbers where the difference between two consecutive numbers is the same, represented by d.

A general form of AP is a, a+d, a+2d…

The nth term of the AP can be calculated using the formula a+(n–1)d.

The chapter ‘Algebra’ is not only an important topic for class 10th mathematics, but it is also an essential concept for advanced mathematics. This chapter’s thorough knowledge will smoothen your journey through the upcoming mathematics chapter, like geometry, equations, number system, etc.

1. What are the different properties of discriminant?
   Following are the different properties of the discriminant of the polynomial:
   D = 0, roots are equal, and real numbers.
   D > 0, roots are distinct and real numbers.
   D < 0, roots are imaginary in nature.
   D = perfect square, roots are distinct and rational numbers.
2. What is the formula to calculate remainder using the remainder theorem?
   If a polynomial is divided by a factor (x – a), then the remainder of the polynomial is f(x).
3. What is the formula to calculate the discriminant or D of quadratic equations?Discriminant (D) of the polynomial is calculated as b^2 – 4ac.
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.