Topics
Number System(Consolidating the Sense of Numberness)
Number System
Theme 1 : Numbers
Number System (Including Estimation and Approximation)
Numbers in Indian and International Systems (With Comparison)
Estimation
Theme 2 : Ratio, Proportion and Arithmetic Problems
Ratio and Proportion
Natural Numbers and Whole Numbers (Including Number Line and Patterns)
Numbers in India and International System (With Comparison)
Algebra
Theme 3 : Algebra
Geometry
Negative Numbers and Integers
Place Value
Theme 4 : Geometry
Theme 5 : Mensuration
Sets
Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Negative Numbers and Integers
Fractions
Theme 6 : Statistics
Data Handling
Number Line
Decimal Fractions
HCF and LCM
Playing with Numbers (Including H.C.F. and L.C.M.)
Playing with Numbers
- Simplification of Brackets
- Finding Factors Using Rectangular Arrangements and Division
- Factors and Common Factors
- Multiples and Common Multiples
- Concept of Even and Odd Number
- Tests for Divisibility of Numbers
- Divisibility by 2
- Divisibility by 4
- Divisibility by 8
- Divisibility by 3
- Divisibility by 6
- Divisibility by 9
- Divisibility by 5
- Divisibility by 11
Ratio and Proportion (Including Unitary Method)
Percent (Percentage)
Sets
Ratio
Idea of Speed, Distance and Time
Proportion (Including Word Problems)
Fundamental Concepts (With operations on Algebraic Expressions)
Unitary Method
Framing Algebraic Expressions (Including Substitution and Linear Equations)
Fractions
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Like and Unlike Fraction
- Concept of Equivalent Fractions
- Conversion between Improper and Mixed fraction
- Conversion between Unlike and Like Fractions
- Simplest Form of a Fractions
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of Fraction
- Division of Fractions
- Using Operator 'Of' with Multiplication and Division
- BODMAS Rule
- Problems Based on Fraction
Fundamental Concepts (Including Angles and their Properties)
Triangles (Including Types and Properties of Triangles)
Decimal Fractions
Quadrilaterals
Percent (Percentage)
Circles
Idea of Speed, Distance and Time
Symmetry
Constructions
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Recognition of Solids
Substitution (Including Use of Brackets as Grouping Symbols)
Perimeter and Area of Plane Figures
Framing Algebraic Expressions (Including Evaluation)
Simple (Linear) Equations (Including Word Problems)
Data Handling (Including Mean and Median)
Fundamental Concepts
Angles (With Their Types)
Properties of Angles and Lines (Including Parallel Lines)
Triangles (Including Types, Properties and Constructions)
Quadrilateral
Polygons
The Circle
Symmetry (Including Constructions on Symmetry)
Recognition of Solids
Perimeter and Area of Plane Figures
Data Handling (Including Pictograph and Bar Graph)
Mean and Median
- Introduction
- Dividing Algebraic Expressions
- Fundamental Rule
- Example: Monomial × Monomial
- Example: Polynomial × Monomial
- Key Points Summary
Introduction
In algebra, we often work with expressions made up of variables and numbers. A monomial is a single term that may include a constant, one or more variables, and their powers — like 7x, −3a2b, or 5xyz2.
Just like numbers can be divided, monomials can also be divided — as long as we follow the laws of exponents. When dividing one monomial by another, we:
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Divide the numerical (constant) parts.
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Subtract the exponents of like variables.
This process helps simplify expressions and is a key skill in algebra, especially when solving equations, simplifying fractions, or working with polynomials.
Dividing Algebraic Expression
Division of a Monomial by a Monomial
- Rule:To divide one monomial by another:
- Divide monomials using basic arithmetic and the quotient rule for exponents.
Division of a Polynomial by a Monomial
- Rule: Divide each term separately by the same monomial.
Fundamental Rule
(i) `(a^m)/(a^n)` = am-n, if m > n and
(ii) `(a^m)/(a^n)` = `(1) / (a^(n − m))` , if n > m
Example: Monomial × Monomial
(i) Division of 12m5 by 4m3 = 12m5 ÷ 4m3
= `( 3 xx \cancel(4) xx \cancel(m) xx \cancel(m) xx \cancel(m) xx m xx m)/(\cancel(4) xx \cancel(m) xx \cancel(m) xx \cancel(m))`
= 3 × m × m = 3m2
(ii) `(x^5y^3)/(x^2y^8)`
= `(x^(5-2))/(y^(8-3))`
= `(x^3)/(y^5)`
Example: Polynomial × Monomial
(i) Division of `(15x^2y^3)` − `(21x^3y^4)` + `(18x^4y^2)` by `(3x^2y^2)`
= `(15x^2y^3)/(3x^2y^2)` − `(21x^3y^4)/(3x^2y^2)` + `(18x^4y^2)/(3x^2y^2)`
= 5y − 7xy² + 6x²
Key Points Summary
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Step 1: Divide the numbers.
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Step 2: Cancel matching letters.
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Step 3: Write what remains.
