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12.2.3 - Factorisation using identities

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Introduction to Factorisation using Identities

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Teacher
Teacher

Today, we will learn how to factor expressions using specific algebraic identities. One important identity is (a + b)² = a² + 2ab + b². Can anyone tell me what this identity means?

Student 1
Student 1

It means that if we have the sum of two terms squared, we can expand it into three terms.

Teacher
Teacher

Exactly! And can someone give me an example of this identity in action?

Student 2
Student 2

What about x² + 6x + 9? It looks like that identity!

Teacher
Teacher

Correct! This can be factored as (x + 3)². Remember, identifying these forms is crucial for simplifying your expressions quickly.

Applying the Identities

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Teacher
Teacher

Now, let’s look at another identity: (a - b)² = a² - 2ab + b². Who can explain when we might use this?

Student 3
Student 3

We would use it when we have a subtraction of two terms squared.

Teacher
Teacher

Right! Can anyone see how this applies to the expression 4y² - 12y + 9?

Student 4
Student 4

Yes! It can be factored down to (2y - 3)².

Teacher
Teacher

Excellent job! By recognizing the form, we can factor a seemingly complex expression. Remember to work through these carefully and look for perfect squares.

Understanding Difference of Squares

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Teacher
Teacher

Now, let’s talk about (a + b)(a - b) = a² - b². How can we apply this? Let’s take 49p² - 36.

Student 1
Student 1

We can rewrite this as (7p)² - (6)², which fits perfectly!

Teacher
Teacher

Exactly! So, it factors to (7p - 6)(7p + 6). Understanding when to apply these identities is key to mastering factorization!

Examples from Class

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Teacher
Teacher

Let’s look at practical examples. For x² + 8x + 16, can we factor it together?

Student 2
Student 2

It’s (x + 4)²!

Teacher
Teacher

Correct! Now, how about 4y² - 12y + 9?

Student 3
Student 3

That would be (2y - 3)².

Teacher
Teacher

Awesome! Remember, practice using these identities, and they will become second nature.

Summarizing the Identities

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Teacher
Teacher

Before we finish, let’s summarize. We learned about three identities: (a + b)², (a - b)², and (a + b)(a - b). Why are these important?

Student 4
Student 4

Because they help us simplify and factor expressions quickly!

Teacher
Teacher

Exactly! Regular practice with these will sharpen your ability to factorise effortlessly. Great work today, everyone!

Introduction & Overview

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

Quick Overview

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

Standard

The section elaborates on three key algebraic identities useful for factorisation: (a + b)², (a - b)², and (a + b)(a - b). It provides examples that guide students on how to identify forms of expressions that match these identities, leading to efficient factorization.

Detailed

In this section, we delve into the process of factorisation using three fundamental identities:

  1. (a + b)² = a² + 2ab + b²
  2. (a - b)² = a² - 2ab + b²
  3. (a + b)(a - b) = a² - b²

Each identity offers a specific framework for recognizing the structure of algebraic expressions and simplifying them accordingly. For instance, when faced with an expression like x² + 8x + 16, we can observe that it aligns with the form (a + b)², allowing us to factor it as (x + 4)². Similarly, expressions such as 49p² - 36 can be simplified using the difference of squares identity. The section is filled with examples to illustrate practical applications of these identities in factorising more complex expressions. By mastering these identities, students can enhance their algebraic skills, making factorisation more intuitive.

Similar Question:

Example : Factorise the expression \( x^2 - 3xy + y^2 - z^2 \).
Solution: The first three terms of the given expression form \( (x - y)^2 \). The fourth term is a square. So the expression can be reduced to a difference of two squares.

Thus,
\[ x^2 - 3xy + y^2 - z^2 = (x - y)^2 - z^2 \]
(Applied Identity II)

\[ = (x - y - z)(x - y + z) \]
(required factorisation)

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Audio Book

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Important Algebraic Identities

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We know that (a + b)² = a² + 2ab + b² (I)
(a - b)² = a² - 2ab + b² (II)
(a + b)(a - b) = a² - b² (III)

Detailed Explanation

This chunk introduces three fundamental identities that are commonly used in factorisation. The first identity, (a + b)² = a² + 2ab + b², represents the square of a binomial, showing how the square of a sum expands. The second identity, (a - b)² = a² - 2ab + b², is similar but applies to the square of a difference. The third identity, (a + b)(a - b) = a² - b², is known as the difference of squares and explains how to factor a difference between two squares. Understanding these identities is crucial for recognizing patterns in polynomial expressions and applying factorisation techniques.

Examples & Analogies

Think of these identities like recipes for baking. Just like combining ingredients in certain proportions gives you a specific dish (like squares or cuboids), these identities allow you to manipulate and transform algebraic expressions in predictable ways. If a recipe calls for sugar and butter to make a cake, you know that changing the amounts will change the cake's flavor and texture. Similarly, knowing how to apply these identities can drastically change how we work with and factor expressions.

Example 1: Factorising x² + 8x + 16

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Observe the expression; it has three terms. Therefore, it does not fit Identity III. Also, its first and third terms are perfect squares with a positive sign before the middle term. So, it is of the form a² + 2ab + b² where a = x and b = 4 such that a² + 2ab + b² = x² + 2(x)(4) + 4². Therefore, x² + 8x + 16 = (x + 4)² (the required factorisation).

Detailed Explanation

In Example 1, we begin by examining the expression x² + 8x + 16. We notice it contains three terms, which makes Identity III inapplicable. However, recognizing that the first term (x²) and the last term (16) are both squares, with the middle term (8x) fitting the form 2ab (where 'a' is x and 'b' is 4), helps us apply Identity I. Consequently, we factor the expression into the form (x + 4)², recognizing that the original expression was a perfect square trinomial.

Examples & Analogies

Imagine you have a square garden measuring x meters on each side. You decide to add a path of 4 meters around the garden. The dimensions change, but the area can be calculated by recognizing that the new dimensions will lead to a larger square garden, represented as (x + 4)². Just as the area can be simplified and factored into manageable parts, so too can the polynomial using identities.

Example 2: Factorising 4y² - 12y + 9

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Observe 4y² = (2y)², 9 = 3² and 12y = 2 × 3 × (2y). Therefore, 4y² - 12y + 9 = (2y)² - 2 × 3 × (2y) + (3)² = (2y - 3)² (required factorisation).

Detailed Explanation

In Example 2, we identify that the expression 4y² - 12y + 9 consists of three terms that can be dissected. By rewriting the terms as squares, we notice that the expression aligns with Identity II, representing the square of a binomial. We proceed to factor the expression as (2y - 3)², revealing it exhibits the characteristics of a perfect square trinomial.

Examples & Analogies

Consider a scenario where you’re designing a small park in a rectangular shape (2y by 2y meters) and want to create a fence (of 3 meters) around it. Just as you can think of the fence as a unique boundary enclosing the garden, when we observe the relationship between these dimensions in our polynomial, it simplifies into a neatly packaged squared term, which helps in understanding space more intuitively.

Example 3: Factorising 49p² - 36

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There are two terms; both are squares and the second is negative. The expression is of the form (a² - b²). Identity III is applicable here; 49p² - 36 = (7p)² - (6)² = (7p - 6)(7p + 6) (required factorisation).

Detailed Explanation

Example 3 illustrates our use of Identity III, applicable when we encounter the difference between two perfect squares: p² and 36. Recognizing that both terms can be expressed as squares enables us to write the expression 49p² - 36 as (7p)² - (6)², subsequently factoring it into (7p - 6)(7p + 6). This format is essential for simplifying and rearranging algebraic expressions.

Examples & Analogies

Think of two friends, each represented by the squares, who are different in size but equally energetic. When they decide to compete, the difference between their energies can be viewed as a product of their unique characteristics, represented algebraically. Hence, observing their strengths can easily translate back into mathematical expressions, allowing us to factorizing supportively, just like understanding their differences leads us to comprehend their competition.

Example 4: Factorising a² - 2ab + b² - c²

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The first three terms of the given expression form (a - b)². The fourth term is a square. So the expression can be reduced to a difference of two squares. Thus, a² - 2ab + b² - c² = (a - b)² - c² = [(a - b) - c)((a - b) + c) (applying Identity III).

Detailed Explanation

In Example 4, we identify that the expression starts with three terms yielding (a - b)². Introducing the last term creates an opportunity for utilizing the difference of squares. We factor the expression as [(a - b) - c] and [(a - b) + c], demonstrating how transformations within algebraic structures can be achieved elegantly by applying the identities appropriately in sequence.

Examples & Analogies

Consider planning an event where two friends (a and b) wish to invite guests from different areas. By plotting their guest lists, you might find overlaps (reflecting the squares) that reveal the need for careful adjustment. Recognizing this, you take away a few invitations (the c term), allowing for a more balanced gathering that factors beautifully into the overall success of the event. This experience of managing relationships resonates with factorization, enabling us to simplify complexities into clearer paths.

Definitions & Key Concepts

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

Key Concepts

  • Factorisation: The process of expressing an expression as a product of its factors.

  • Identities: Specific mathematical statements that hold true and can be applied to simplify expressions.

  • Perfect Square Trinomials: Quadratic expressions of the form (a + b)² or (a - b)² that can be factored using identities.

Examples & Real-Life Applications

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

Examples

  • Example 1: Factor x² + 8x + 16 as (x + 4)².

  • Example 2: Factor 4y² - 12y + 9 as (2y - 3)².

  • Example 3: Factor 49p² - 36 as (7p - 6)(7p + 6).

Memory Aids

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

🎵 Rhymes Time

  • To factor and not to fear, perfect squares will appear!

📖 Fascinating Stories

  • Once in Algebra Land, identities were the heroes who helped the students factor all their difficult problems with ease!

🧠 Other Memory Gems

  • Remember 'PES' for Perfect Squares: Perfect = (a + b)² or (a - b)², Exponential = a² or b², Simplify the expression!

🎯 Super Acronyms

FIP for Factor in pieces

  • F: = Factor
  • I: = Identify form
  • P: = Perfect squares.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: (a + b)²

    Definition:

    An algebraic identity that expands as a² + 2ab + b².

  • Term: (a b)²

    Definition:

    An algebraic identity that expands as a² - 2ab + b².

  • Term: Difference of Squares

    Definition:

    An expression of the form (a + b)(a - b) that simplifies to a² - b².

  • Term: Factorisation

    Definition:

    The process of breaking down an expression into products of simpler factors.

  • Term: Perfect Square

    Definition:

    An expression that can be written as the square of another expression.