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Introduction to Factorisation
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Good morning, class! Today, we start talking about factorisation. Who can tell me what factorisation might mean?
Isn't it when we break something down into parts or factors?
Absolutely! Factorisation is all about expressing an expression as a product of factors. Can anyone give me an example of a simple factorisation?
Like 2x + 4 can be factored into 2(x + 2)?
Great example, Student_2! So remember, factorisation can help simplify expressions and solve equations. Let's keep this in mind! Does anyone need clarification?
Common Factor Method
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Now, letβs discuss one technique for factorisation: the common factor method. Can someone explain how we could use this with the expression 12aΒ²b + 15abΒ²?
We could find the common factors! I think 3ab is common in both terms.
Perfect! So we can factor it as 3ab(4a + 5b). Thatβs what we aim for! Who can tell me why identifying the common factor is important?
Because it simplifies the expression and makes it easier to work with!
Exactly! Keeping things simpler helps in solving problems faster.
Regrouping for Factorisation
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Letβs move on to regrouping. This is particularly useful when no single factor is shared. For example, in 2xy + 2y + 3x + 3. What can we do here?
We could group the first two terms and the last two terms together!
Right! So we can write it as (2y(x + 1) + 3(x + 1)). Notice what's common now?
We can factor out (x + 1) again!
Exactly! So, we have (x + 1)(2y + 3). Keep practicing regrouping; it can lead to efficient factorisation.
Using Identities for Factorisation
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Now, we will talk about factorisation using identities. For example, can anyone tell me how we can use the identity for (a + b)Β²?
I remember that it expands to aΒ² + 2ab + bΒ². We can also do the reverse, right?
Exactly! So, if given xΒ² + 8x + 16, we can factor it as (x + 4)Β². Whatβs important here is to recognize these forms quickly.
So, we watch for squares and other familiar forms in our expressions?
You got it! This recognition will help you greatly in factorisation, leading to quicker solutions.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we explore the concept of factorisation in algebra, learning how to express algebraic expressions as products of their factors. We cover various techniques such as finding common factors, regrouping terms, and utilizing identities for factorisation, providing practical examples and exercises to reinforce these methods.
Detailed
What is Factorisation?
Factorisation refers to the process of expressing an algebraic expression as a product of its factors. Factors can be numerical values, algebraic variables, or entire algebraic expressions themselves. Some expressions are already in their factor form like 3xy or 5(x+1)(x+2), while others, such as 2x + 4, require techniques to identify their factors.
Key Techniques of Factorisation:
- Method of Common Factors: This involves identifying the highest common factor (HCF) in terms of numerical coefficients and algebraic variables and then factoring it out of the expression.
- Regrouping Terms: In some cases, it's beneficial to rearrange or regroup the terms in an expression to isolate common factors. This method allows us to factor expressions that do not have an obvious common factor across all terms.
- Using Identities: Familiarity with algebraic identities, such as
- (a + b)Β² = aΒ² + 2ab + bΒ²
- (a - b)Β² = aΒ² - 2ab + bΒ²
- aΒ² - bΒ² = (a + b)(a - b)
can facilitate the factorisation of expressions fitting these forms.
We also discuss more complicated expressions like quadratics and polynomials, illustrating methods to factor them systematically. Through practice and exercises, learners gain familiarity with these techniques, essential for solving various algebraic problems.
Similar Question:
Example 2: Factorise
\[ 18ac + 24a^2c^2 \]
Solution:
We have
\[ 18ac = 2 \times 3 \times 3 \times a \times c \]
\[ 24a^2c^2 = 2 \times 2 \times 2 \times 3 \times a \times a \times c \times c \]
The two terms have \( 6ac \) as a common factor.
Therefore,
\[ 18ac + 24a^2c^2 = 6ac \left( 3 + 4ac \right) \]
\[ = 6ac(3 + 4ac) \quad (\text{required factor form}) \]
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Understanding 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.
Detailed Explanation
Factorisation involves rewriting an expression in the form of a product of its factors. For example, the expression 12 can be factorised into 3 Γ 4. In algebra, we can do this with variables too, such as writing xy as x Γ y. The key idea here is that factorisation expresses an expression in a simpler form.
Examples & Analogies
Think of factorisation like breaking down a complex recipe into its individual ingredients. Just as a cake can be expressed as flour, eggs, and sugar, algebraic expressions can be expressed in simpler terms (factors).
Identifying Expressions in Factor Form
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Expressions like 3xy, 5x2y, 2x(y + 2), 5(y + 1)(x + 2) are already in factor form. Their factors can be just read off from them, as we already know.
Detailed Explanation
Certain expressions are already presented as products of factors. For instance, 3xy is already in factorised form because it's expressed as the product of 3, x, and y. Recognising these expressions makes it easier to work with them because we can directly see the factors involved.
Examples & Analogies
Imagine you have a box of toys. If the box contains three types of toys - cars, dolls, and blocks -, you can think of the box as being in 'factor form' by directly counting how many of each toy you have (the factors) instead of listing each toy one by one.
Finding the Factors of More Complex Expressions
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On the other hand consider expressions like 2x + 4, 3x + 3y, x2 + 5x, x2 + 5x + 6. It is not obvious what their factors are. We need to develop systematic methods to factorise these expressions, i.e., to find their factors.
Detailed Explanation
Some expressions are not immediately clear in terms of their factors. For example, with 2x + 4, we need to identify common elements and break it down into its simplest factor form. This becomes necessary for the expressions where visualising or seeing factors is not straightforward, requiring techniques such as factoring by grouping or using identities.
Examples & Analogies
Think about a puzzle that is all mixed up. At first glance, you see a jumble of pieces (the expression) and it is not clear what the final picture (the factors) will be. You need to sort pieces into groups (factorisation techniques) to successfully put the puzzle together again.
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 Γ x, 4 = 2 Γ 2.
Detailed Explanation
To factor the expression 2x + 4, we break each term down. We note that both terms share a common factor: 2. By factoring out 2, we can rewrite the expression as 2(x + 2). This process helps in simplifying expressions and solving equations more easily.
Examples & Analogies
Consider gathering all your red apples and green apples into different baskets. If you have 2 apples in one basket and 4 apples in another basket with the same color, you can say you have a common factor of apples that you can put together to show a group of 2 apples, making it visual and easier to manage.
Example of Factorisation
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Example 1: Factorise 12a2b + 15ab2
Detailed Explanation
First, we find the irreducible factors of each term. 12a2b can be broken down to 2 Γ 2 Γ 3 Γ a Γ a Γ b and 15ab2 as 3 Γ 5 Γ a Γ b Γ b. The common factors are 3ab. We can then factor the expression as 3ab(4a + 5b). This clearly shows how we can write complex algebraic expressions in simpler factorised terms.
Examples & Analogies
Imagine packing boxes for a party. If you have a total of 12 cupcakes and 15 cookies, you can group them in twos to make organizing and serving easier, simplifying a later task in the event just as factorisation simplifies mathematical expressions.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Factorisation: Process of expressing an algebraic expression as a product of its factors.
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Common Factors: Identifying shared factors in terms to simplify expressions.
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Regrouping: Rearranging terms to reveal common factors for factorisation.
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Algebraic Identities: Special forms that can be used to simplify factorisation.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example: Factor 12aΒ²b + 15abΒ² as 3ab(4a + 5b).
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Example: Regroup 2xy + 2y + 3x + 3 to (x + 1)(2y + 3).
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Example: Use of identity to factor xΒ² + 8x + 16 as (x + 4)Β².
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Factor out the greatest with ease, common terms will surely please!
π Fascinating Stories
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Imagine a farmer with various baskets of fruits, each with some common fruits; he groups them by types to get easier access.
π§ Other Memory Gems
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C.R.A.F.T. - Common factors, Regrouping, Algebraic identities For easy Technique.
π― Super Acronyms
F.A.C.T. - Factor, Algebraic expressions, Common factors, Techniques.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Factorisation
Definition:
The process of writing an algebraic expression as a product of its factors.
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Term: Common Factor
Definition:
A number or variable that divides two or more terms evenly.
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Term: Regrouping
Definition:
The rearranging of terms in an expression to facilitate factorisation.
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Term: Irreducible Factor
Definition:
A factor that cannot be further expressed as a product of factors.
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Term: Algebraic Identities
Definition:
Equations that are true for all values of the variables involved.