Topics
Rational Numbers
- Rational Numbers
- Closure Property of Rational Numbers
- Commutative Property of Rational Numbers
- Associative Property of Rational Numbers
- Distributive Property of Multiplication Over Addition for Rational Numbers
- Identity of Addition and Multiplication of Rational Numbers
- Negative Or Additive Inverse of Rational Numbers
- Concept of Reciprocals or Multiplicative Inverses
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
- Multiples and Common Multiples
Linear Equations in One Variable
- Constants and Variables in Mathematics
- Equation in Mathematics
- Expressions with Variables
- Word Problems on Linear Equations
- Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Solving Equations Having the Variable on Both Sides
- Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
- Reducing Equations to Simpler Form
- Equations Reducible to Linear Equations
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Basic Concept of Polygons
- Classification of Polygons
- Properties of Quadrilateral
- Sum of Interior Angles of a Polygon
- Sum of Exterior Angles of a Polygon
- Quadrilaterals
- Properties of Trapezium
- Properties of Kite
- Properties of a Parallelogram
- Properties of Rhombus
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The adjacent angles in a parallelogram are supplementary.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Properties of Rectangle
- Properties of a Square
- Property: The diagonals of a square are perpendicular bisectors of each other.
Data Handling
Practical Geometry
- Geometric Tool
- Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
- Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
- Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
- Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
- Some Special Cases
Squares and Square Roots
- Concept of Square Number
- Properties of Square Numbers
- Some More Interesting Patterns of Square Number
- Finding the Square of a Number
- Concept of Square Roots
- Finding Square Root Through Repeated Subtraction
- Finding Square Root Through Prime Factorisation
- Finding Square Root by Division Method
- Square Root of Decimal Numbers
- Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
- Ratio
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Calculation of Interest
- Concept of Compound Interest
- Deducing a Formula for Compound Interest
- Rate Compounded Annually Or Half Yearly (Semi Annually)
- Applications of Compound Interest Formula
Algebraic Expressions and Identities
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Classification of Terms in Algebra
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Monomial by a Trinomial
- Multiplying a Binomial by a Binomial
- Multiplying a Binomial by a Trinomial
- Concept of Identity
- Expansion of (a + b)2 = a2 + 2ab + b2
- Expansion of (a - b)2 = a2 - 2ab + b2
- Expansion of (a + b)(a - b) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Exponents and Powers
Visualizing Solid Shapes
Direct and Inverse Proportions
Factorization
- Factors and Common Factors
- Factorising Algebraic Expressions
- Factorisation by Taking Out Common Factors
- Factorisation by Regrouping Terms
- Factorisation Using Identities
- Factors of the Form (x + a)(x + b)
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Concept of Find the Error
Introduction to Graphs
Playing with Numbers
- Introduction
- Multiplying Algebraic Expressions
- Examples: Monomial × Monomial
- Examples: Monomial × Polynomial
- Examples: Binomial × Binomial
- Key Points Summary
Introduction
Multiplication of algebraic expressions involves multiplying the terms of the given polynomials or monomials. The basic rule for multiplying powers of like bases is to add the exponents. Additionally, we can multiply monomials with polynomials, or two binomials, using distributive properties.
Multiplying Algebraic Expressions
Multiplying Monomials
- Rule: Multiply coefficients, then add exponents of like variables.
- Formula: am × an = am+n
Multiplying a Monomial by a Polynomial
- Rule: Distribute the monomial across each term of the polynomial.
- Multiply term-by-term, then simplify.
Multiplying Two Binomials
- Rule: Multiply each term of the first binomial by each term of the other one.
- In the product obtained, combine the like terms
Examples: Monomial × Monomial
(i) x5 × x7
= x5+7 = x12.
(ii) x2y3 × x6y4
= x2+6 y3 + 4 = x8y7
(iii) xy2z5 × x4y3z2 × x7y5z4
= x1+4+7 × y2+3+5 × z5 + 2 + 4
= x12y10z11
Examples: Monomial × Polynomial
(i) Multiplication of 4x²y − 3xy² + 4xy and 2xy
= 2xy × (4x²y − 3xy² + 4xy)
= (2xy × 4x²y) − (2xy × 3xy²) + (2xy × 4xy)
= 8x³y² − 6x²y³ + 8x²y²
(ii) 5x² (3x³y² − 2xy³ + y⁵)
= (5x² × 3x³y²) − (5x² × 2xy³) + (5x² × y⁵)
= 15x⁵y² − 10x³y³ + 5x²y⁵
Examples: Binomial × Binomial
(i) Multiplication of x + 3 and x + 5
= (x + 3) . (x + 5)
= x . (x + 5) + 3 . (x + 5)
= x.x + x.5 + 3.x + 15
= x² + 5x + 3x + 15
= x² + 8x + 15
(ii) Multiply: 3x − 4y and 4x + 5x.
(3x − 4y) · (4x + 5y)
= 3x(4x + 5y) − 4y(4x + 5y)
= 12x² + 15xy − 16xy − 20y²
= 12x² − xy − 20y²
Alternative method:
3x − 4y
× 4x + 5y
12x² − 16xy [4x. (3x − 4y) = 12x² − 16xy]
+ 15xy − 20y² [5y. (3x - 4y) = 15xy − 20y²]
12x² − xy − 20y²
12x² − xy − 20y²
Key Points Summary
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Monomials: Multiply coefficients, add exponents of like variables.
-
Monomial × Polynomial: Distribute the single term across each term, then simplify.
-
Binomials:Multiply each term in the first binomial with each term in the second binomial, then combine the like terms.
