Factorisation Exercise 14.2 Class 8 NCERT: Concepts & Solutions

Core Concepts: Reverse Application of Identities

To factorise expressions in Exercise 14.2, we look at the expression and match it with one of our standard algebraic products. Essentially, factorisation is the reverse process of multiplication. If we know that multiplying $(a+b)$ by itself gives $a^2 + 2ab + b^2$, then factorising $a^2 + 2ab + b^2$ must return us to $(a+b)^2$. The three main identities we use here are the square of a sum, the square of a difference, and the difference of two squares. When you see a trinomial, always check if the first and last terms are perfect squares. If they are, it is highly likely you can use the identity $(a+b)^2$ or $(a-b)^2$. If you see a binomial with a minus sign between two squared terms, you can instantly apply the difference of squares identity.

Key Identities to Memorize

Identity I (Perfect Square Sum)

a² + 2ab + b² = (a + b)²

Identity II (Perfect Square Difference)

a² - 2ab + b² = (a - b)³

Identity III (Difference of Squares)

a² - b² = (a - b)(a + b)

Identity IV (General Trinomial)

x² + (a + b)x + ab = (x + a)(x + b)

How to Factorise Step-by-Step

1. Look for Common Factors First — Before applying any identity, examine all the terms in the algebraic expression. If there is a common numeric or variable factor across all terms, factor it out first.
2. Identify the Structure of the Expression — Count the number of terms. If there are 3 terms, check if it matches Identity I, II, or IV. If there are 2 terms with a subtraction sign, check if it matches Identity III.
3. Express Terms as Perfect Squares — Rewrite the appropriate terms as perfect squares of simple monomials. For instance, write 16x² as (4x)².
4. Substitute and Write the Final Factors — Apply the matching identity to write the polynomial in its factorised product form.

NCERT-Style Solved Examples

• Example 1: Factorise 4y² - 12y + 9. Solution: Here, the first term is (2y)² and the last term is 3². The middle term is -12y, which can be written as -2(2y)(3). Since it matches the form a² - 2ab + b², we use Identity II. Thus, 4y² - 12y + 9 = (2y - 3)².
• Example 2: Factorise 49p² - 36. Solution: This expression has two terms separated by a minus sign. Rewrite both as perfect squares: (7p)² - 6². Using Identity III, a² - b² = (a - b)(a + b), we get (7p - 6)(7p + 6).

Exam Tips & Pitfalls to Avoid

1. Watch the sign of the middle term: When working with three terms, if the middle term is negative, ensure you use the $(a-b)^2$ identity and not $(a+b)^2$.
2. Incomplete Factorisation: Always check if your resulting brackets can be factorised further. For example, $(x^4 - y^4)$ factorises to $(x^2 - y^2)(x^2 + y^2)$, but $(x^2 - y^2)$ must be factorised further to $(x - y)(x + y)$.
3. Don't forget to take out common factors first: If you try to factorise $2x^2 - 18$ directly, you might get stuck. If you take out the common factor $2$ first, it becomes $2(x^2 - 9) = 2(x-3)(x+3)$.

Practice Questions with Solutions

• Q: Factorise the algebraic expression: a² + 8a + 16 A: Step 1: Identify the perfect squares. The first term is a² = (a)². The third term is 16 = (4)². Step 2: Check the middle term. It is 8a, which equals 2 × (a) × (4). This fits the pattern a² + 2ab + b². Step 3: Apply the identity (a + b)². Here, a is 'a' and b is '4'. Final answer: (a + 4)²
• Q: Factorise the expression: 4x² - 25y² A: Step 1: Identify the structure. It is a difference of two terms, both of which can be written as perfect squares. Step 2: Rewrite each term as a square: 4x² = (2x)² and 25y² = (5y)². Step 3: Apply Identity III: a² - b² = (a - b)(a + b), where a = 2x and b = 5y. Final answer: (2x - 5y)(2x + 5y)
• Q: Factorise the trinomial: x² + 5x + 6 A: Step 1: This trinomial does not have a perfect square as its constant term (6 is not a perfect square). We must find two numbers whose sum is 5 and product is 6. Step 2: The numbers are 2 and 3, since 2 + 3 = 5 and 2 × 3 = 6. Step 3: Split the middle term and group: x² + 2x + 3x + 6 = x(x + 2) + 3(x + 2). Step 4: Take out the common binomial factor (x + 2). Final answer: (x + 2)(x + 3)
• Q: Factorise the expression: l² - 2lm + m² - z² A: Step 1: Notice that the first three terms l² - 2lm + m² form a perfect square. This can be simplified using Identity II to (l - m)². Step 2: Rewrite the entire expression as (l - m)² - z². Step 3: This is now a difference of two squares. Apply Identity III where a = (l - m) and b = z. Final answer: (l - m - z)(l - m + z)

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