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12 - Factorisation

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Listen to a student-teacher conversation explaining the topic in a relatable way.

Understanding Factors

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Teacher
Teacher

Today, we're going to explore what factors are! Who can tell me what they think factors are?

Student 1
Student 1

I think factors are numbers that can multiply together to get another number.

Teacher
Teacher

Exactly! For example, 30 can be broken down as 2 times 15. And what about its prime factors?

Student 2
Student 2

A prime factor is a factor that is a prime number, right? Like 2, 3, or 5 for 30?

Teacher
Teacher

Great! So the prime factors of 30 are 2, 3, and 5. Remember, every number has at least two factors: itself and 1. We'll use this knowledge in factorisation.

Teacher
Teacher

Can anyone tell me why we write 30 as a product of its prime factors?

Student 3
Student 3

It's because it makes it easier to see how the number is built up!

Teacher
Teacher

Exactly! So remember: factors and prime factors help us in breaking down and understanding numbers.

Teacher
Teacher

To remember this, think of the acronym 'FARM' — Factors Are Really Magnificent!

Algebraic Expressions and Irreducible Factors

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Teacher
Teacher

Moving on, let's talk about algebraic expressions. Who can give me an example of an algebraic expression?

Student 4
Student 4

Like 5xy + 3x?

Teacher
Teacher

Exactly! And can you identify the factors in the term 5xy?

Student 1
Student 1

The factors are 5, x, and y!

Teacher
Teacher

Correct! We call these 'irreducible' factors. They can’t be simplified further. What's another example?

Student 2
Student 2

3x(x + 2)!

Teacher
Teacher

Perfect! Now, to find the factors of more complex expressions, what strategy do we use?

Student 3
Student 3

We look for common factors first!

Teacher
Teacher

Exactly! Remember the phrase 'Check Commonly!' to remind yourself.

Factoring Techniques

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Teacher
Teacher

Let's dive into methods of factorisation. Who remembers the first technique we discussed?

Student 4
Student 4

Finding the common factors!

Teacher
Teacher

That's right! Let's factor 2x + 4 together.

Student 1
Student 1

2x = 2 × x and 4 = 2 × 2, so it's like 2(x + 2)!

Teacher
Teacher

Correct! Now, what if we don't have a common factor across all terms?

Student 2
Student 2

We can regroup the terms!

Teacher
Teacher

Exactly! Regrouping allows us to create groups that share common factors. Now let's try that with the expression 2xy + 2y + 3x + 3.

Student 3
Student 3

We can group 2xy + 2y together and 3x + 3 together!

Teacher
Teacher

Absolutely! And when we factor out, we end up with (x + 1)(2y + 3). Knocking down expressions, one factor at a time — remember 'KNOCK!'

Factoring Using Identities

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Teacher
Teacher

Now, let's explore factorisation using identities. For instance, can someone tell me a useful identity?

Student 3
Student 3

Yeah! (a + b)² = a² + 2ab + b²!

Teacher
Teacher

Exactly! And we can recognize this form in expressions? For example, x² + 8x + 16.

Student 1
Student 1

It fits the identity for a perfect square!

Teacher
Teacher

Perfect! When we see such patterns, we can factor using the identities we've learned. Can anyone tell me how this can simplify expressions?

Student 4
Student 4

It makes it much easier to factor without expanding them manually!

Teacher
Teacher

Wonderful! Remember the identity is like a magic key, allowing you to open doors to easier factorisation. Just think 'MAGIC KEY!'

Division of Algebraic Expressions

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Teacher
Teacher

Let's switch gears to division of algebraic expressions. Can someone summarize how division relates to what we've learned about multiplication?

Student 2
Student 2

Division is the opposite of multiplication!

Teacher
Teacher

Exactly! For example, if I take 6x³ and want to divide it by 2x, what would that look like?

Student 3
Student 3

It simplifies to 3x²!

Teacher
Teacher

Right you are! We can simplify the expression based on common factors. What's important to note here?

Student 4
Student 4

We can use factorisation to make division easier!

Teacher
Teacher

Yes! So remember this: 'Facts to Simplify!' when dividing.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

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.

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Audio Book

Dive deep into the subject with an immersive audiobook experience.

Introduction to Factors

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You will remember what you learnt about factors in Class VI. Let us take a natural number, say 30, and write it as a product of other natural numbers, say 30 = 2 × 15. We know that 30 can also be written as 3 × 10 = 5 × 6 = 1 × 30. Thus, 1 and 30 are also factors of 30. Thus, 1, 2, 3, 5, 6, 10, 15 and 30 are the factors of 30. You will notice that 1 is a factor of any number. A number written as a product of prime factors is said to be in the prime factor form; for example, 30 written as 2 × 3 × 5 is in the prime factor form.

Detailed Explanation

In mathematics, a factor is a natural number that divides another number completely without leaving a remainder. Here, we used the number 30 as an example to illustrate this. We can express 30 in several ways by multiplying pairs of numbers that give 30 when multiplied together. For instance, 2 × 15 and 3 × 10 both give us 30. Collectively, all the numbers that can multiply together to give 30 (which are called factors) are 1, 2, 3, 5, 6, 10, 15, and 30. It's also important to note that every number has 1 as a factor, making it a universal factor.

Examples & Analogies

Think of factors like friends of a number. Just as you can have a group of friends (like 1, 2, 3, etc.) who can come together or pair up (like in the pairs (2, 15) or (3, 10)) to achieve a common goal (in this case, the number 30), factors work similarly for numbers.

Prime Factorisation

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Of these, 2, 3 and 5 are the prime factors of 30 (Why?). A number written as a product of prime factors is said to be in the prime factor form. The prime factor form of 70 is 2 × 5 × 7. The prime factor form of 90 is 2 × 3 × 3 × 5, and so on.

Detailed Explanation

The prime factors of a number are the prime numbers that multiply together to give the original number. In the case of our example with 30, the prime factors are 2, 3, and 5, since these cannot be divided any further into smaller natural numbers that are also prime. When we break numbers down into their prime components, we say we are finding the prime factorization. This process helps simplify calculations in various mathematical operations.

Examples & Analogies

Think of prime factors like the basic building blocks of a number. Just as you can build a house using basic materials like bricks (where each brick represents a prime factor), you can reconstruct any number using its prime factors.

Factors in Algebra

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Similarly, we can express algebraic expressions as products of their factors. This is what we shall learn to do in this chapter. We have seen in Class VII that in algebraic expressions, terms are formed as products of factors. For example, in the algebraic expression 5xy + 3x, the term 5xy has been formed by the factors 5, x and y.

Detailed Explanation

Algebraic expressions, just like natural numbers, can also be broken down into factors. In the algebraic expression 5xy + 3x, you can identify that 5xy consists of three factors: 5, x, and y. This process of breaking down expressions into simpler multiplicative components is known as factorization, which we will extensively cover in this lesson. Understanding the factors of algebraic expressions is crucial as it lays the foundation for simplification, solving equations, and other advanced mathematical concepts.

Examples & Analogies

Consider the expression 5xy as a recipe where 5 is the quantity of a specific ingredient, and xy is a combination of two flavors. Just like you can alter the recipe by changing the quantities of the ingredients while keeping the flavor constant, you can rearrange or factor out parts of algebraic expressions.

What is Factorisation?

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When we factorise an algebraic expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions. Expressions like 3xy, 5x2y, 2x(y + 2) are already in factor form. On the other hand consider expressions like 2x + 4, 3x + 3y, x2 + 5x.

Detailed Explanation

Factorisation refers to the process of breaking down an expression into products of other factors. While some expressions, like 3xy and 5x²y, are already in their factor form, others, like 2x + 4, need to be factorized to reveal their underlying structure. This is important as it allows for easier manipulation, simplification, and understanding of the expression's nature.

Examples & Analogies

Imagine factorisation as taking apart a toy set to reveal its individual pieces. While some toys come pre-assembled (like expressions in factor form), others need to be taken apart and reassembled to see how they fit together.

Method of Common Factors

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We begin with a simple example: Factorise 2x + 4. We shall write each term as a product of irreducible factors; 2x = 2 × x and 4 =2 × 2. Hence 2x + 4 =(2 × x) + (2 × 2). Notice that factor 2 is common to both the terms.

Detailed Explanation

To factorize an expression effectively, we can start by identifying common factors among the terms. In our example with 2x + 4, both terms share a factor of 2. By factoring out the common factor, we arrive at the simplified expression 2(x + 2). This method of identifying common factors can be applied to a wide variety of algebraic expressions to simplify them efficiently.

Examples & Analogies

Consider a scenario where you have multiple boxes filled with the same type of item (like cookies). Instead of counting each box separately, you can take out the common boxes and focus on the number of boxes remaining, making it easier to manage the total number.

Factorisation by Regrouping Terms

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Look at the expression 2xy + 2y + 3x + 3. You will notice that the first two terms have common factors 2 and y and the last two terms have a common factor 3. But there is no single factor common to all the terms.

Detailed Explanation

In more complex expressions, we may find that no one factor is common across all terms. In such cases, we can regroup the terms into groups that do share common factors. For example, in the expression 2xy + 2y + 3x + 3, we can regroup to find common factors: the first group can be factored as 2y(x + 1), and the second group as 3(x + 1). This leads to the final factorization of (x + 1)(2y + 3).

Examples & Analogies

Think of this process like organizing a messy room. If you look for items scattered all around, it may be daunting. However, if you start collecting similar items together (like clothes, toys, etc.), it becomes more manageable to tidy them up.

Factorisation Using Identities

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We know that (a + b)² = a² + 2ab + b². The following solved examples illustrate how to use these identities for factorisation.

Detailed Explanation

Mathematic identities provide powerful tools to expedite the factorization process. For instance, the identity for the square of a sum, (a + b)² = a² + 2ab + b², allows us to identify patterns within expressions that can be expressed similarly. If we see an expression resembling this pattern, we can use the identity to factor it easily and accurately.

Examples & Analogies

Think of using a recipe for making a cake. If you know the ingredients needed for a basic cake (like flour, eggs, and sugar), you can easily substitute those for a fancier cake without needing to figure everything out from scratch.

Factorisation of Quadratic Expressions

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Consider the expression x² + 5x + 6. We compare the expression with the identity (x + a)(x + b). We find that ab = 6, and a + b = 5. From this, we must obtain a and b.

Detailed Explanation

When we deal with quadratic expressions, we can express them in terms of two factors rather than a standard polynomial. By analyzing a quadratic expression like x² + 5x + 6, we recognize it can often be factored into (x + a)(x + b). Here, finding factors a and b that multiply to the constant term (6) and add to the coefficient of the linear term (5) allows us to simplify the expression further.

Examples & Analogies

Think of this like combining different fruits into a fruit salad. You need to find the right balance of flavors (the ‘a’ and ‘b’ factors) to make it delicious while representing the overall quantity (the quadratic expression).

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

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.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

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

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 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.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for 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).