12. Factorisation

12. Factorisation

  • 12

    Factorisation

    This section covers the concept of factorisation, emphasizing how numbers and algebraic expressions can be expressed as products of factors.

  • 12.1

    Introduction

    This section introduces the concepts of factorization in natural numbers and algebraic expressions, explaining factors, prime factors, and their applications.

  • 12.1.1

    Factors Of Natural Numbers

    This section introduces factors of natural numbers, illustrating their identification and importance in mathematics.

  • 12.1.2

    Factors Of Algebraic Expressions

    This section explores the concept of factors in algebraic expressions and different methods of factorisation.

  • 12.2

    What Is Factorisation?

    Factorisation is the process of writing an algebraic expression as a product of its factors, which may include numbers, variables, or other expressions.

  • 12.2.1

    Method Of Common Factors

    This section introduces the method of factorization using common factors by breaking down algebraic expressions into irreducible components.

  • 12.2.2

    Factorisation By Regrouping Terms

    This section teaches the method of factorisation by regrouping terms in algebraic expressions, highlighting the process through specific examples.

  • 12.2.3

    Factorisation Using Identities

    This section introduces factorisation through algebraic identities, demonstrating how to recognize and apply these identities to simplify expressions.

  • 12.2.4

    Factors Of The Form (X + A)(X + B)

    This section discusses how to factor expressions in one variable, specifically those that can be expressed in the form of (xxxxx + aaaaa)(xxxxx + bbbbb), including strategies to find the correct factors.

  • 12.3

    Division Of Algebraic Expressions

    This section introduces the concept of dividing algebraic expressions, focusing on the division of monomials and polynomials.

  • 12.3.1

    Division Of A Monomial By Another Monomial

    This section explains how to divide a monomial by another monomial using factorization.

  • 12.3.2

    Division Of A Polynomial By A Monomial

    This section explains how to perform division of a polynomial by a monomial, highlighting common factors and simplification methods.

  • 12.4

    Division Of Algebraic Expressions Continued

    This section discusses the division of polynomials by polynomials and focuses on factorization techniques for algebraic expressions.

    • Key Summary

    The chapter covers the concepts of factorization including the identification of factors of natural numbers and algebraic expressions. It explains various methods for factorizing algebraic expressions through common factors, regrouping, and identities. The division of algebraic expressions is also introduced, presenting methods for dividing monomials and polynomials.

    Key Takeaways

    • When we factorise an expression, we write it as a product of factors including numbers and algebraic variables or expressions.
    • An irreducible factor cannot be expressed further as a product of factors.
    • A systematic approach, such as the common factor method, is essential for accurate factorization.

    Key Concepts

    • Factorization: Writing an expression as a product of its factors, which may include numbers, algebraic variables, or expressions.
    • Common Factor Method: A systematic method of factorization involving identifying and separating common factors from terms.
    • Regrouping: Rearranging terms in an expression to facilitate factoring by identifying common factors within groups of terms.
    • Polynomial Division: The process of dividing a polynomial by another polynomial or monomial, which may involve cancellation of common factors.