# Chapter 2 – Algebra

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

Introduction

ICSE Class 10th Mathematics chapter ‘Algebra‘ is an essential chapter as it forms the basis of advanced mathematics. It covers all the vital topics related to algebra, geometry, and equations. It is also used in successive chapters of class 10th mathematics, so a clear understanding of algebra will better understand other mathematical concepts.

Chapter ‘Algebra’ covers the following topics:

• Introduction to algebra
• Linear inequalities
• Factorisation of polynomials
• Quadratic equations in one variable
• Ratio and proportion
• Matrices
• Coordinate geometry
• Arithmetic progression

#### Introduction to Algebra

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.

#### Linear Inequalities

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.

#### Factorisation of Polynomials

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 in One Variable

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.

#### Ratio and Proportion

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.

#### Matrices

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.

#### Coordinate Geometry

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 formula of 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

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.

#### Conclusion

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.

#### FAQs

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.

‘Algebra‘ is an important chapter of mathematics. So, the visual representation of this chapter is essential. MSVgo offers a large base of learning videos and interactive learning methods like quizzes, interactive games, and more. MSVgo is a smartphone application available on both android and IOS that explains every concept with interactive visual representations. Check out the MSVgo application to know more and get access to explanatory science and mathematics videos.

### 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
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• Introduction to Organic Chemistry
• Ionic equilibria
• Matter
• Matter Around Us
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• Metallurgy
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• 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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