NCERT Solutions for Class 8 Maths Chapter 12 Factorisation

What is Factorisation?

Factorisation is the process of expressing an algebraic expression as a product of factors. A factor is an algebraic expression, a number, or a variable. For instance, consider an algebraic expression 3x + 15. This can be factorised as 3(x + 5). Here, 3 and x + 5 are factors of the given algebraic expression.

Class 8 factorisation mainly uses three methods:

• Common Factor Method: Take out the highest common factor from all terms.

• Regrouping Method: Rearrange terms to spot common factors in pairs.

• Identity Method: Use algebraic identities to split expressions instantly.

Chapter Overview & Important Topics

Chapter 12 focuses on factorising algebraic expressions using methods that make problem-solving simpler. Key topics include:

• Identifying common factors

• Factorisation by regrouping

• Using algebraic identities

• Handling quadratic expressions

• Division of polynomials

This overview ensures you know what to focus on for your exercises and exams.

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Factorisation Class 8 NCERT Solutions

Finding reliable solutions is key to mastering this chapter. The factorisation class 8 question answer guide provides clear, step-by-step logic for every exercise. We break down complex polynomial divisions and identity applications so you can verify your homework and prepare for exams with complete confidence.

Exercise 12.1 – Common Factors

Find the common factors: 12x, 36

• Sol: 12x = 2 × 2 × 3 × x, 36 = 2 × 2 × 3 × 3. The common numerical factors are 2, 2, and 3.

• Ans: 12

Find the common factors: 14pq, 28p²q²

• Sol: 14pq = 2 × 7 × p × q. 28p²q² = 2 × 2 × 7 × p² × q². Common = 14pq. In a problem involving variables like pk, you would look for shared letters in the same way.

• Ans: 14pq

Read More - NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

Exercise 12.2 – Factorisation by Grouping

Factorise: 7x – 42

• Sol: Both terms divide by 7. Pull 7 out of the brackets.

• Ans: 7(x – 6)

Factorise: 5x²y – 15xy²

• Sol: The common factor is 5xy. If the expression used pk instead, like 5pk – 15p²k, the common factor would be 5pk.

• Ans: 5xy(x – 3y)

Exercise 12.3 – Factorisation using Identities & Division

Factorise: a² + 8a + 16

• Sol: Matches a² + 2ab + b² → (a + 4)².

• Ans: (a + 4)²

Divide: (5x² – 6x) ÷ 3x

• Sol: Factor the numerator x(5x – 6), then cancel x. If we had a numerator like 3pk(x+1) divided by pk, the pk terms would simply cancel out.

• Ans: (5x – 6)/3

Exercise 12.4 - What are the Four Main Methods of Factorization?

The Four Main Methods of Factorisation are:

To solve any factorisation class 8 question answer, you need to pick the right tool for the job. Most problems fall into one of these four categories:

This is the most basic starting point. You look at every term in the expression and find the Highest Common Factor (HCF) of the numerical coefficients and the variables. Once you find it, you pull it outside a bracket.

Factorisation by Regrouping

Sometimes, an expression has four terms, and no single factor is common to all of them. In this case, we group the terms into pairs. We factorise each pair separately, which often reveals a new common binomial factor.

Using Algebraic Identities

Golden identities, in short, were generally derived from the difference of squares and perfect squares. Golden identities are shortcuts that enable you to skip the step of manually splitting and jump to the answer directly.

Splitting the Middle Term

For quadratic expressions in the form x^2 + (a + b)x + ab, we find two numbers that add up to the middle coefficient and multiply to the last term.

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Important Identities & Formulae in Factorization

Keep these identities handy while factorising. They save time and reduce errors during high-pressure tests:

• (a + b)² = a² + 2ab + b²

• (a − b)² = a² − 2ab + b²

• a² − b² = (a + b)(a − b)

• x² + (a + b)x + ab = (x + a)(x + b)

How to Check Your Factorisation Results?

Use these verification steps to ensure you get every class 8 chapter 12 maths question answer correct.

1. The Expansion Method: Multiply your factors back together. If they don't match the original exactly, check your signs.

2. Observe the Signs: Pulling a negative out of brackets flips the internal signs.

3. Variable Count: If you start with a term like pk, ensure both p and k are accounted for in your final factors.

4. Logical Check: The Logical Check requires you to extract any remaining common factor from your last brackets before completing the HCF examination.

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