Factorisation

12 Factorisation

Description

Quick Overview

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

Standard

The section explores the definitions of factors of natural numbers and algebraic expressions, different methods of factorisation including common factors and regrouping, and introduces several identities useful for factorisation. It also provides methods for dividing algebraic expressions.

Detailed

Factorisation Summary

In this section, we learn about factorisation, specifically how both natural numbers and algebraic expressions can be expressed as products of their factors.

Factors of Natural Numbers

We recall the definition of factors, with examples using numbers like 30 (whose factors include 1, 2, 3, 5, 6, 10, 15, and 30). We highlight prime factors, explaining that a number expressed as a product of prime factors is in its prime factor form (

Factors of Algebraic Expressions

In algebra, terms are products of factors; for instance, in the expression 5xy + 3x, the term 5xy consists of the factors 5, x, and y. We redefine 'prime' factors as 'irreducible' factors in terms of algebra.

What is Factorisation?

Factorisation is explained as writing an algebraic expression or number as the product of factors, distinguishing between expressions already in factor form and those requiring methods to factor them systematically.

Methods of Factorisation

  1. Common Factors: The method begins with identifying common factors in terms and using the distributive property to rewrite expressions.
  2. Regrouping: This involves rearranging terms to identify common factors when there is no single common factor across the entire expression.

Factorisation Using Identities

This section also introduces various identities (like the square of a binomial and the difference of squares) which assist in factorising specific expressions.

Division of Algebraic Expressions

Finally, we cover how to divide algebraic expressions, including the division of monomials and polynomials by monomials and other polynomials, emphasizing that we can factor out common terms and simplify expressions accordingly.

Key Concepts

  • Factors: Numbers or expressions multiplied to yield another number or expression.

  • Irreducible Factors: The simplest form of factors that cannot be simplified further.

  • Factorisation: The method of expressing expressions or numbers as products of their factors.

  • Common Factors: A shared factor amongst given terms used in factorization.

  • Regrouping: A strategic rearrangement of terms to facilitate the identification of common factors.

  • Identities: Fundamental equations that help in simplifying and solving expressions.

Memory Aids

🎵 Rhymes Time

  • Factors come in pairs, big and small, When multiplied together, they answer the call!

📖 Fascinating Stories

  • Once there was a number named 30 who had many friends—1, 2, 3, all the way to 30! They would all gather and multiply to create new friends.

🧠 Other Memory Gems

  • To remember factor pairs can use the phrase 'Pull and Pair!'—pull out common factors and pair the others.

🎯 Super Acronyms

F.A.C.T.S. — Factors Are Crucial To Simplifying.

Examples

    1. Factorising 12a²b + 15ab²: This results in 3ab(4a + 5b).
    1. Applying identities, we factor x² + 8x + 16 as (x + 4)².

Glossary of Terms

  • Term: Factors

    Definition:

    Numbers or expressions that are multiplied together to get another number or expression.

  • Term: Irreducible Factors

    Definition:

    Factors that cannot be expressed as a product of further simpler factors.

  • Term: Prime Factors

    Definition:

    Factors that are prime numbers, meaning they only have two distinct positive divisors: 1 and themselves.

  • Term: Factorisation

    Definition:

    The process of expressing a number or algebraic expression as a product of its factors.

  • Term: Common Factor

    Definition:

    A number or expression that divides two or more numbers or terms evenly.

  • Term: Regrouping

    Definition:

    A method of arranging terms in an expression to find common factors.

  • Term: Identities

    Definition:

    Equations that are true for all values of the variable(s) involved.

  • Term: Difference of Squares

    Definition:

    An algebraic expression in the form a² - b², factorable into (a + b)(a - b).