What is Factorisation by Grouping?

S6-SA1-0052

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Factorisation by Grouping is a method to break down algebraic expressions into simpler parts (factors) when they have four or more terms. We group terms that share a common factor, then factor out that common factor from each group, eventually finding a common binomial factor.

Simple Example

Quick Example

Imagine you have a big basket of fruits: 3 apples, 6 bananas, 5 oranges, and 10 grapes. You can group them! You could group (3 apples + 6 bananas) because both numbers are divisible by 3. And (5 oranges + 10 grapes) because both numbers are divisible by 5. This makes it easier to count or share. Factorisation by Grouping does something similar with numbers and variables in math.

Worked Example

Step-by-Step

Let's factorise the expression: 2xy + 3x + 4y + 6

Step 1: Group terms that share common factors. Look at the first two terms and the last two terms. (2xy + 3x) and (4y + 6).
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Step 2: Factor out the common factor from the first group. In (2xy + 3x), 'x' is common. So, x(2y + 3).
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Step 3: Factor out the common factor from the second group. In (4y + 6), '2' is common. So, 2(2y + 3).
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Step 4: Now the expression looks like: x(2y + 3) + 2(2y + 3). Notice that (2y + 3) is common to both terms.
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Step 5: Factor out the common binomial factor (2y + 3). We get (2y + 3)(x + 2).
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Answer: The factorised form is (2y + 3)(x + 2).

Why It Matters

Factorisation by Grouping is crucial for simplifying complex equations, which is super important in fields like Engineering and Physics to solve problems quickly. It helps AI/ML algorithms process data more efficiently and is used in Biotechnology to model molecular interactions. Understanding this helps you build a strong foundation for future careers in technology and science.

Common Mistakes

MISTAKE: Not finding a common binomial factor after the first step of grouping. For example, getting x(a+b) + 2(c+d). | CORRECTION: If the binomials aren't the same, try grouping the terms in a different way or check your common factors.

MISTAKE: Forgetting to include the sign when taking out a common factor, especially a negative one. For example, factorising -4y - 6 as -2(2y - 3). | CORRECTION: Always double-check the signs. -4y - 6 should be -2(2y + 3) because -2 * 3 = -6.

MISTAKE: Thinking that if there are no common factors in the original expression, it cannot be factorised. | CORRECTION: Factorisation by Grouping works by finding common factors *within* groups, even if the entire expression doesn't have one overall common factor.

Practice Questions

Try It Yourself

QUESTION: Factorise: ab + 5a + 3b + 15 | ANSWER: (b + 5)(a + 3)

QUESTION: Factorise: x^2 - 7x + 3x - 21 | ANSWER: (x - 7)(x + 3)

QUESTION: Factorise: 6mn - 4m + 3n - 2 | ANSWER: (3n - 2)(2m + 1)

MCQ

Quick Quiz

Which of the following is the correct factorisation of 5xy - 10x + 3y - 6?

(y - 2)(5x + 3)

(y + 2)(5x - 3)

(y - 2)(5x - 3)

(y + 2)(5x + 3)

The Correct Answer Is:

Group (5xy - 10x) as 5x(y - 2) and (3y - 6) as 3(y - 2). Both groups share (y - 2), so factor it out to get (y - 2)(5x + 3).

Real World Connection

In the Real World

Engineers use factorisation to simplify equations when designing bridges or calculating how much load a building can handle. For example, when creating a new app for ordering food like Zomato or Swiggy, developers might use factorisation in the backend code to optimise calculations for delivery routes or pricing algorithms, making the app run faster and smoother for users across India.

Key Vocabulary

Key Terms

FACTOR: A number or expression that divides another number or expression exactly | EXPRESSION: A combination of numbers, variables, and operation symbols | BINOMIAL: An algebraic expression with two terms | COMMON FACTOR: A factor that two or more numbers or expressions share

What's Next

What to Learn Next

Great job understanding Factorisation by Grouping! Next, you should learn about 'Factorisation of Quadratic Polynomials'. This will build on your grouping skills to factorise expressions with x^2 terms, which are very common in higher math and science.