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Chapter 2 – Algebra

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

Introduction

The most basic fundamentals in mathematics is the equation. An equation is a mathematical statement consisting of an equal sign in between two mathematical expressions, where both sides of the equation can be rearranged or manipulated (by applying mathematical operations) in order to solve the equation.

Algebra is nothing but these equations. A combination of consonants and variables connected by the signs of fundamental operations of addition, subtraction, multiplication, and division, which is called an Algebraic Expression. Algebra can include real numbers, complex numbers, matrices, vectors, and many more forms of mathematical representation. No matter if you love Algebra or it’s a bane of your existence, you must learn to solve Algebraic Expressions. It has got many real-life examples; a few of them include – Health & Fitness, Logical Thinking, Technology, Business Management, and Professional Advancement.

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 –

  •  a2 – b2 = (a – b)(a + b)
  •  (a+b)2 = a2 + 2ab + b2
  •  a2 + b2 = (a – b)2 + 2ab
  •  (a – b)2 = a2 – 2ab + b2
  •  (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
  •  (a – b – c)2 = a2 + b2 + c2 – 2ab – 2ac + 2bc
  •  (a + b)3 = a3 + 3a2b + 3ab2 + b3
  •  (a – b)3 = a3 – 3a2b + 3ab2 – b3

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:

    am X an = am+n

    Eg – 32  X 33 = 35

2. Dividing powers with the same base:

    am ÷ an = am-n

    Eg – 24 ÷ 22 = 22

3. Taking powers of powers :

    ( am )n = am X n

    Eg – ( 23)2 = 26

4. Multiplying powers with the same exponents:

    am X bm= (ab)m

    Eg – 23 X 33 = 63

The multiplication of rational numbers possesses the following properties –

  1. Commutativity – The multiplication of rational numbers is commutative. That is, if ab and cd are any two fractions then, ab * cd= cd * ab 
  2. Associativity: the multiplication of rational numbers is associative. That is, if ab ,cd , ef are three fractions , then ( ab * cd )*ef = ab *( cd * ef )
  3. 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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High School Physics

  • Alternating Current
  • Atoms
  • Communication Systems
  • Current Electricity
  • Dual nature of Radiation and Matter
  • Electric Charges and Fields
  • Electricity
  • Electromagnetic Induction
  • Electromagnetic Waves
  • Electron Beams and Radioactivity
  • Electrons and Photons
  • Electrostatic Potential and Capacitance
  • Fluid Pressure
  • Force and Acceleration
  • Force And Laws Of Motion
  • Gravitation
  • Internal Energy
  • Kinetic Theory
  • Law of motion
  • Light – Reflection And Refraction
  • Magnetic Effects Of Electric Current
  • Magnetism and Matter
  • Management Of Natural Resources
  • Mechanical properties of Fluids
  • Mechanical properties of Solids
  • Motion
  • Motion in a plane
  • Motion in a straight line
  • Moving Charges and Magnetism
  • Nuclear Energy
  • Nuclei
  • Oscillations
  • Our Environment
  • Paths of Heat
  • Physical world
  • Ray optics and optical instruments
  • Semiconductor Devices
  • Semiconductor Electronics: Materials, Devices and Simple Circuits
  • Simple Machines
  • Sound
  • Sources Of Energy
  • Specific and Latent Heats
  • Spherical Mirrors
  • Static Electricity
  • Systems of Particles and Rotational motion
  • Thermal properties of matter
  • Thermodynamics
  • Units and Measurement
  • Vectors, Scalar Quantities and Elementary Calculus
  • Wave Optics
  • Waves
  • Work, Power and Energy

High School Chemistry

  • Acids, Bases and Salts
  • Alcohols, Phenols and Ethers
  • Aldehydes, Ketones and Carboxylic Acids
  • Aliphatic and Aromatic Hydrocarbons
  • Alkyl and Aryl Halides
  • Amines
  • Analytical Chemistry 
  • Atomic Structure
  • Atoms And Molecules
  • Basic concepts of Chemistry
  • Biomolecules
  • Carbon And Its Compounds
  • Carboxylic acids and Acid Derivatives
  • Chemical Bonding and Molecular Structures
  • Chemical Energetics
  • Chemical Equilibria
  • Chemical Kinetics
  • Chemical Reactions And Equations
  • Chemical Reactions and Their Mechanisms
  • Chemistry in Everyday Life
  • Chemistry of p-Block elements
  • Chemistry of Transition and Inner Transition
  • Classification of Elements
  • Coordination Compounds
  • Cyanide, Isocyanide, Nitro compounds and Amines
  • Electrochemistry
  • Electrolysis
  • Elements, Compounds and Mixtures
  • Environmental Chemistry
  • Equilibrium
  • Ethers and Carbonyl compounds
  • Haloalkanes and Haloarenes
  • Hydrocarbons
  • Hydrogen
  • Ideal solutions
  • Introduction to Organic Chemistry
  • Ionic equilibria
  • Matter
  • Matter Around Us
  • Matter In Our Surroundings
  • Metallurgy
  • Metals And Non-Metals
  • Mole Concept and Stoichiometry
  • Natural Resources
  • Organic Chemistry – Basic Principles
  • Periodic Classification of Elements
  • Physical and Chemical Changes
  • Physical and Chemical Properties of Water
  • Polymers
  • Preparation, Properties and Uses of Compounds
  • Principles and Processes of Isolation of Elements
  • Redox Reactions
  • Relative Molecular Mass and Mole
  • States of Matter
  • Structure Of The Atom
  • Study of Compounds
  • Study of Gas Laws
  • Study of Representative Elements
  • Surface Chemistry
  • The d-block and f-block elements
  • The Gaseous State
  • The p-Block Elements
  • The Periodic Table
  • The s-Block Elements
  • The Solid State
  • Thermodynamics

High School Biology

  • Absorption and Movement of Water in Plants
  • Adolescent Issues
  • Anatomy of Flowering Plants
  • Animal Kingdom
  • Bacteria and Fungi-Friends and Foe
  • Biodiversity and Conservation
  • Biofertilizers
  • Biological Classification
  • Biomedical Engineering
  • Biomolecules
  • Biotechnology and its Applications
  • Biotic Community
  • Body Fluids and Circulation
  • Breathing and Exchange of Gases
  • Cell – Unit of Life
  • Cell Cycle and Cell Division
  • Cell Division and Structure of Chromosomes
  • Cell Reproduction
  • Cellular Respiration
  • Chemical Coordination and Integration
  • Circulation
  • Control And Coordination
  • Crop Improvement
  • Digestion and Absorption
  • Diversity In Living Organisms
  • Ecosystem
  • Environmental Issues
  • Excretory Products and their Elimination
  • Flowering Plants
  • Genes and Chromosomes
  • Health and Diseases
  • Health and Its Significance
  • Heredity And Evolution
  • Heredity and Variation
  • How Do Organisms Reproduce?
  • Human Diseases
  • Human Eye And Colourful World
  • Human Health and Disease
  • Human Population
  • Human Reproduction
  • Hygiene
  • Improvement In Food Resources
  • Integumentary System- Skin
  • Kingdom Fungi
  • Kingdom Monera
  • Kingdom Protista
  • Life Processes
  • Locomotion and Movement
  • Microbes in Human Welfare
  • Mineral Nutrition
  • Molecular Basis of Inheritance
  • Morphology of Flowering Plants
  • Neural Control And Coordination
  • Nutrition in Human Beings
  • Organism and Population
  • Photosynthesis
  • Photosynthesis in Higher Plants
  • Plant Growth and Development
  • Plant Kingdom
  • Pollination and Fertilization
  • Pollution; Sources and its effects
  • Principles of Inheritance and Variation
  • Reproduction and Development in Angiosperms
  • Reproduction in Organisms
  • Reproductive Health
  • Respiration in Human Beings
  • Respiration in Plants
  • Respiratory System
  • Sexual Reproduction in Flowering Plants
  • Strategies for Enhancement in Food Production
  • Structural Organisation in Animals
  • Structural Organisation of the Cell
  • The Endocrine System
  • The Fundamental Unit Of Life
  • The Living World
  • The Nervous System and Sense Organs
  • Tissues
  • Transpiration
  • Transport in Plants

High School Math

  • Algebra – Arithmatic Progressions
  • Algebra – Complex Numbers and Quadratic Equations
  • Algebra – Linear Inequalities
  • Algebra – Pair of Linear Equations in Two Variables
  • Algebra – Polynomials
  • Algebra – Principle of Mathematical Induction
  • Algebra – Quadratic Equations
  • Binomial Theorem
  • Calculus – Applications of Derivatives
  • Calculus – Applications of the Integrals
  • Calculus – Continuity and Differentiability
  • Calculus – Differential Equations
  • Calculus – Integrals
  • Geometry – Area
  • Geometry – Circles
  • Geometry – Conic Sections
  • Geometry – Constructions
  • Geometry – Introduction to Euclid’s Geometry
  • Geometry – Three-dimensional Geometry
  • Geometry – Lines and Angles
  • Geometry – Quadrilaterals
  • Geometry – Straight Lines
  • Geometry – Triangles
  • Linear Programming
  • Matrices and Determinants
  • Mensuration – Areas
  • Mensuration – Surface Areas and Volumes
  • Number Systems
  • Number Systems – Real Numbers
  • Permutations and Combinations
  • Probability
  • Sequence and Series
  • Sets and Functions
  • Statistics 
  • Trignometry – Height and Distance
  • Trignometry – Identities
  • Trignometry – Introduction

Middle School Science

  • Acids, Bases And Salts
  • Air and Its Constituents
  • Basic Biology
  • Body Movements
  • Carbon and Its Compounds
  • Cell – Structure And Functions
  • Changes Around Us
  • Chemical Effects Of Electric Current
  • Chemistry in Your Life
  • Coal And Petroleum
  • Combustion And Flame
  • Components Of Food
  • Conservation Of Plants And Animals
  • Crop Production And Management
  • Electric Current And Its Effects
  • Electricity And Circuits
  • Elements and Compounds
  • Fibre To Fabric
  • Food production and management
  • Force And Pressure
  • Forests: Our Lifeline
  • Friction
  • Fun With Magnets
  • Garbage In, Garbage Out
  • Getting To Know Plants
  • Health and Hygiene
  • Heat
  • Hydrogen
  • Life Processes: Nutrition in Animals and Plants
  • Light, Shadows And Reflections
  • Materials: Metals And Non-Metals
  • Matter and Its States
  • Metals and Non-metals
  • Micro Organisms: Friend And Foe
  • Motion And Measurement Of Distances
  • Motion And Time
  • Nutrition In Animals
  • Nutrition In Plants
  • Organization in Living Things
  • Our Environment
  • Physical And Chemical Changes
  • Pollution and conservation
  • Pollution Of Air And Water
  • Reaching The Age Of Adolescence
  • Reproduction In Animals
  • Reproduction In Plants
  • Respiration In Organisms
  • Rocks and Minerals
  • Separation Of Substances
  • Simple Machines
  • Soil
  • Some Natural Phenomena
  • Sorting Materials Into Groups
  • Sound
  • Stars And The Solar System
  • Structure of Atom
  • Synthetic Fibers And Plastics
  • The Living Organisms And Their Surroundings
  • Transfer of Heat
  • Transformation of Substances
  • Transportation In Animals And Plants
  • Universe
  • Waste-water Story
  • Water: A Precious Resource
  • Weather, Climate And Adaptations Of Animals To Climate
  • Winds, Storms And Cyclones

Middle School Math

  • Addition
  • Area and Its Boundary
  • Boxes and Sketches
  • Data Handling
  • Fun With Numbers
  • Heavy and Light
  • How Many
  • Long And Short
  • Mapping
  • Measurement
  • Money
  • Multiplication and Factors
  • Multiply and Divide
  • Numbers
  • Parts and Wholes
  • Pattern Recognition
  • Patterns
  • Play With Patterns
  • Rupees And Paise
  • Shapes And Angles
  • Shapes And Designs
  • Shapes and Space
  • Similarity
  • Smart Charts
  • Squares
  • Subtraction
  • Tables And Shares
  • Tenths and Hundredths
  • Time
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