What is Factorisation of Algebraic Expressions?

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Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Factorisation of algebraic expressions is like breaking down a number into its prime factors, but for expressions with variables. It's the process of writing an algebraic expression as a product of two or more simpler expressions (called factors). Think of it as finding the 'ingredients' that multiply together to make the original expression.

Simple Example

Quick Example

Imagine you have a rectangle whose area is given by the expression '3x + 6'. We want to find its length and breadth. Factorising '3x + 6' means we are finding two expressions that multiply to give '3x + 6'. Here, we can write it as '3(x + 2)'. So, '3' and '(x + 2)' are the factors.

Worked Example

Step-by-Step

Let's factorise the algebraic expression: 4y + 12

1. Identify the terms in the expression: The terms are 4y and 12.
---2. Find the greatest common factor (GCF) of the numerical coefficients (4 and 12): The factors of 4 are 1, 2, 4. The factors of 12 are 1, 2, 3, 4, 6, 12. The greatest common factor is 4.
---3. Check if there's a common variable: In this case, 'y' is only in the first term, so there's no common variable.
---4. The GCF of the entire expression is 4.
---5. Write the GCF outside a bracket: 4( )
---6. Divide each original term by the GCF: (4y / 4) = y and (12 / 4) = 3.
---7. Write these results inside the bracket: 4(y + 3).
---8. The factorised expression is 4(y + 3). You can check by multiplying: 4 * y + 4 * 3 = 4y + 12. This matches the original expression.

Answer: 4(y + 3)

Why It Matters

Factorisation is super important in fields like Computer Science and Engineering for solving complex equations and optimising code. It helps AI/ML models process data efficiently and is even used in Cryptography to secure your online transactions. Mastering it opens doors to exciting careers in technology and research.

Common Mistakes

MISTAKE: Not finding the *greatest* common factor (GCF). For example, factorising 6x + 12 as 2(3x + 6) | CORRECTION: Always find the largest number or variable that divides all terms. The correct factorisation for 6x + 12 is 6(x + 2).

MISTAKE: Forgetting to divide *all* terms by the common factor. For example, factorising 5a + 10 as 5a + 2 | CORRECTION: Remember to divide every single term inside the expression by the common factor you've taken out. It should be 5(a + 2).

MISTAKE: Confusing factorisation with simplification. For example, changing 2x + 4 to 6x | CORRECTION: Factorisation is about rewriting an expression as a product of factors, not combining terms. Simplification is combining like terms. For 2x + 4, factorisation is 2(x + 2).

Practice Questions

Try It Yourself

QUESTION: Factorise the expression: 7m + 21 | ANSWER: 7(m + 3)

QUESTION: Factorise the expression: 9p - 18q | ANSWER: 9(p - 2q)

QUESTION: Factorise the expression: 5x + 10y + 15z | ANSWER: 5(x + 2y + 3z)

MCQ

Quick Quiz

Which of the following is the correct factorisation of 8a + 16?

2(4a + 8)

8(a + 2)

4(2a + 4)

8a + 16

The Correct Answer Is:

The greatest common factor of 8a and 16 is 8. Dividing both terms by 8 gives (8a/8) = a and (16/8) = 2. So, the correct factorisation is 8(a + 2). Options A and C use common factors but not the *greatest* one, and Option D is the original expression, not factorised.

Real World Connection

In the Real World

Think about designing a cricket stadium. Engineers use factorisation to simplify equations for calculating material requirements or optimising seating arrangements. If the area of a section is an algebraic expression, factorising it helps them find suitable dimensions (length and width) for different parts, making the construction process more efficient and cost-effective.

Key Vocabulary

Key Terms

FACTOR: An expression that divides another expression exactly | COMMON FACTOR: A factor that is shared by two or more terms in an expression | GREATEST COMMON FACTOR (GCF): The largest common factor among terms | TERM: A single number, variable, or product of numbers and variables in an expression (e.g., 3x, 5, 2y^2)

What's Next

What to Learn Next

Great job learning the basics of factorisation! Next, you can explore 'Factorisation by Grouping' and 'Factorisation using Identities'. These concepts build on finding common factors and will help you factorise even more complex expressions, preparing you for higher-level algebra.