What is Factorisation by Regrouping?

S3-SA1-0826

Grade Level:

Class 6

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

Definition

What is it?

Factorisation by regrouping is a method used to break down a polynomial (an expression with many terms) into simpler parts, called factors. We do this by rearranging the terms and finding common factors within smaller groups of terms. It's like sorting your toys into groups before packing them.

Simple Example

Quick Example

Imagine you have 4 friends, and each friend has a cricket bat and a ball. You could write this as (Bat + Ball) + (Bat + Ball) + (Bat + Ball) + (Bat + Ball). Or, you could say 4 x (Bat + Ball). Here, 4 is a common factor. Factorisation by regrouping helps when the items are mixed up, like if some friends have only bats and others have only balls, and you need to group them smartly.

Worked Example

Step-by-Step

Let's factorise the expression: 2x + 6y + ax + 3ay

STEP 1: Look at the terms and try to find pairs that might have common factors. Let's group (2x + 6y) and (ax + 3ay).
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STEP 2: Factor out the common factor from the first group (2x + 6y). The common factor is 2. So, 2(x + 3y).
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STEP 3: Factor out the common factor from the second group (ax + 3ay). The common factor is 'a'. So, a(x + 3y).
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STEP 4: Now our expression looks like: 2(x + 3y) + a(x + 3y).
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STEP 5: Notice that (x + 3y) is a common factor in both terms. Factor it out.
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STEP 6: So, we get (x + 3y)(2 + a).
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ANSWER: The factorised expression is (x + 3y)(2 + a).

Why It Matters

Factorisation by regrouping is a fundamental skill used in many advanced fields. Engineers use it to simplify complex equations for designing bridges or circuits. In Computer Science, it helps in optimising algorithms for faster calculations. It's a stepping stone for understanding how complex systems are broken down into manageable parts.

Common Mistakes

MISTAKE: Not identifying the correct common factor in each group, leading to different terms inside the brackets. For example, in 2x + 6y, taking out 'x' instead of '2'. | CORRECTION: Always find the Greatest Common Factor (GCF) for each group of terms.

MISTAKE: Not getting the same bracketed term after factoring each group. For example, ending up with (x+3y) and (x-3y). | CORRECTION: If the bracketed terms are not identical, re-check your grouping or your common factor extraction. You might need to rearrange the original terms or factor out a negative sign.

MISTAKE: Forgetting to write the final factored form as a product of two factors, like (common bracket)(remaining terms). For example, just writing '2(x+3y) + a(x+3y)' as the answer. | CORRECTION: Remember the goal is to express the entire polynomial as a product of its factors. The final step is to factor out the common bracket.

Practice Questions

Try It Yourself

QUESTION: Factorise: 3a + 3b + xa + xb | ANSWER: (a + b)(3 + x)

QUESTION: Factorise: 5p - 10q + mp - 2mq | ANSWER: (p - 2q)(5 + m)

QUESTION: Factorise: x^2 + 2xy - 3xz - 6yz | ANSWER: (x + 2y)(x - 3z)

MCQ

Quick Quiz

Which of these expressions is the correct factorisation of 4a + 4b + ca + cb by regrouping?

(4 + c)(a - b)

(4 - c)(a + b)

(a + b)(4 + c)

(a - b)(4 + c)

The Correct Answer Is:

By grouping (4a + 4b) and (ca + cb), we get 4(a + b) + c(a + b). Factoring out (a + b) gives (a + b)(4 + c). Options A, B, and D have incorrect signs or groupings.

Real World Connection

In the Real World

Imagine you are a shopkeeper managing inventory for a small electronics store in Delhi. You have a stock of phone chargers and headphones, some for Android phones and some for iPhones. If you list your inventory as 'Android Chargers + Android Headphones + iPhone Chargers + iPhone Headphones', factorisation by regrouping helps you quickly see the total number of Android accessories (Android(Chargers + Headphones)) and iPhone accessories (iPhone(Chargers + Headphones)), making stock management easier. This is similar to how supply chain logistics use such thinking to group items efficiently.

Key Vocabulary

Key Terms

FACTOR: A number or expression that divides another number or expression exactly. | TERM: A single number or variable, or numbers and variables multiplied together. | POLYNOMIAL: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. | COMMON FACTOR: A factor that two or more numbers or expressions share.

What's Next

What to Learn Next

Great job understanding factorisation by regrouping! Next, you should explore 'Factorisation by using Identities'. This will show you even quicker ways to factorise special types of expressions, building on the grouping skills you just learned.