##### Chapter 2

## Algebra

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

#### Introduction

Algebra is taught in schools all over India, beginning between the sixth and eighth grades and continuing well into high school and even college. It also has heavy marks weightage.

Terminology Used in Algebra

**Variable**: A symbol that takes various numerical values is called a variable. It is generally in the form of letters.

**Constant**: Numerical value attached with a variable, which is fixed always is known as consonants.

In the equation 2x – 9, 2 is constant, and x is variable.

**Terms**: Various parts of an algebraic expression that are separated by the signs of + or – are called “terms” of the expression.

**Monomial:** An algebraic expression containing only one term is known as Monomial. For example, 3x, 8y, 7z.

**Binomial**: An algebraic expression containing two terms is called a binomial. For example, 7x – 8y, 9x – 8z.

**Trinomial**: An algebraic expression containing three terms is called a trinomial. For example, 3x 2 + 7x + 1.

**Coefficient:** In terms of algebraic expression, any of the factors with the sign of the term is called the coefficient of the product of the other factors. For example, in – 5xy, the coefficient of x is – 5y.

**Like and Unlike terms:** The terms having the same literal factors are called ‘like’ or ‘similar terms’; otherwise, they are called, ‘unlike’ terms. Example: In the expression 2a – 3b + 5a – 6b, 2a and 5a are like terms, and the rest are unlike terms.

Basic Algebra identities include –

- a
^{2}– b^{2}= (a – b)(a + b) - (a+b)
^{2}= a^{2}+ 2ab + b^{2} - a
^{2}+ b^{2}= (a – b)^{2}+ 2ab - (a – b)
^{2}= a^{2}– 2ab + b^{2} - (a + b + c)
^{2}= a^{2}+ b^{2}+ c^{2}+ 2ab + 2ac + 2bc - (a – b – c)
^{2}= a^{2}+ b^{2}+ c^{2}– 2ab – 2ac + 2bc - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - (a – b)
^{3}= a^{3}– 3a^{2}b + 3ab^{2}– b^{3}

While solving algebraic equations, we follow the BODMAS Rule. It defines the order of operations where terms inside Bracket are solved first, then orders (power, root), multiplication is done next, and lastly, we do addition and subtraction.

We can write large numbers in a short form using Exponents.

For any non-zero rational number ‘a’ and a natural n, the product a x a x a x a x a…x a i.e., the continued product of ‘a’ multiplied with itself n-times, is written as an. It is known as the nth power of ‘a’ and is read as ‘a’ raised to the power ‘n.’ The rational number ‘a’ is called the base, and ‘n’ is called the exponent or index.

This notation of writing the product of a rational number by itself several times is called the Exponential Notation or Power Notation.

Negative integral exponent – For any non-zero rational number ‘a’ and a positive integer, we define a – n = i.e, a – n is the reciprocal of a n

1. Multiplying powers with the same base:

a^{m} X a^{n} = a^{m+n}

Eg – 3^{2 } X 3^{3} = 3^{5}

2. Dividing powers with the same base:

a^{m} ÷ a^{n} = a^{m-n}

Eg – 2^{4} ÷ 2^{2} = 2^{2}

3. Taking powers of powers :

( a^{m} )n = a^{m X n}

Eg – ( 2^{3})^{2} = 2^{6}

4. Multiplying powers with the same exponents:

a^{m} X b^{m}= (ab)^{m}

Eg – 2^{3} X 3^{3} = 6^{3}

The multiplication of rational numbers possesses the following properties –

- Commutativity – The multiplication of rational numbers is commutative. That is, if ab and cd are any two fractions then, ab * cd= cd * ab
- Associativity: the multiplication of rational numbers is associative. That is, if ab ,cd , ef are three fractions , then ( ab * cd )*ef = ab *( cd * ef )
- Distributivity of multiplication over addition: the multiplication of rational numbers is distributive over their addition. That is if ab, cd, ef are three rational numbers, then

ab * ( cd + ef ) = ab * cd + ab * ef

In the multiplication of algebraic expressions, we shall be using the following rules of signs:

(i) The product of two factors with like signs is positive, and the product of two factors with unlike signs is negative.

(ii) If a is any variable and m, n are positive integers, then am x an = am + n

The coefficient and variable part in the product of two or more monomials are equal to the product of the coefficient and variable parts, respectively, in the given monomials.

Multiplication of a Monomial and a Binomial – By using the distributivity of multiplication over addition property, we multiply a monomial and a binomial.

Thus, if P, Q, and R are three monomials, then we have –

(i) P X (Q + R) = (P X Q) + (P X R)

(ii) (Q + R) X P = (Q X P) – (R X P)

In the multiplication of two binomials, we use the distributive property of multiplication over addition.

Consider two binomials, say (a+b) and (c+d).By using the property we have –

(a + b) x (c + d) = a x (c + d) + b x (c + d)

= (a x c + a x d) + (b x c + b x d)

= ac + ad + bc + bd

1) What are the basics of algebra?

At the Elementary level, Algebra includes-

- Addition and subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Division of Algebraic Expressions
- Solving Equations using substitution or elimination method

2) What are the branches of algebra?

The branches of algebra include –

- Pre- Algebra
**Elementary Algebra****Advanced Algebra****Abstract Algebra****Linear Algebra**- Universal Algebra

3) What are the three branches of mathematics?

The three branches of mathematics are –

- Algebra
- Trigonometry
- Geometry
- Calculus
- Statistics and Probability

4) Who is the father of mathematics?

Archimedes, the Greek mathematician, is regarded as the Father of Mathematics for his notable mathematics and science achievements.

5) Who is the No. 1 mathematician in the world?

Pythagoras, who gave us the right-angled triangle, is the No.1 mathematician of the world.

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