Class 8 — Mathematics (NCERT)

Factorisation

Class 8

✓By the end of this lesson students will be able to understand the meaning of factorisation of algebraic expressions.
✓By the end of this lesson students will be able to factorise algebraic expressions by taking out common factors.
✓By the end of this lesson students will be able to factorise algebraic expressions by grouping terms.
✓By the end of this lesson students will be able to factorise algebraic expressions using standard algebraic identities.
✓By the end of this lesson students will be able to perform division of algebraic expressions.

Key concepts

What is Factorisation?

Factorisation is the process of expressing an algebraic expression as a product of two or more expressions. These expressions are called the factors of the given algebraic expression. It is the reverse process of multiplication. For example, just as 12 can be written as 2 × 2 × 3, an algebraic expression like 3x + 9 can be written as 3(x + 3), where 3 and (x + 3) are the factors.

Method of Common Factors

In this method, we look for factors that are common to all terms in the given algebraic expression. We find the Highest Common Factor (HCF) of the numerical coefficients and the common literal factors (variables) with the lowest power present in all terms. Then, we write the expression as the product of the common factor and the remaining expression obtained by dividing each term by the common factor.

ab + ac = a(b + c)

Factorisation by Grouping Terms

This method is used when there is no common factor among all the terms of an expression. We group the terms in such a way that each group has a common factor. After factorising each group, we should find a common binomial factor, which can then be taken out to complete the factorisation.

ax + bx + ay + by = x(a + b) + y(a + b) = (a + b)(x + y)

Factorisation using Identities

Some algebraic expressions can be factorised by recognising them as the R.H.S. of standard algebraic identities. The three main identities used for factorisation at this level are:

I. (a + b)² = a² + 2ab + b²\nII. (a - b)² = a² - 2ab + b²\nIII. a² - b² = (a - b)(a + b)

Division of Algebraic Expressions

Division of algebraic expressions involves finding the quotient when one expression (dividend) is divided by another (divisor). We can perform division in different scenarios:\n\n1. **Division of a Monomial by a Monomial**: Divide the numerical coefficients and subtract the powers of the same variables.\n Example: 12x³y² ÷ 3xy = (12/3) * (x³/x) * (y²/y) = 4x²y\n\n2. **Division of a Polynomial by a Monomial**: Divide each term of the polynomial by the monomial separately.\n Example: (6x³ + 9x²) ÷ 3x = (6x³/3x) + (9x²/3x) = 2x² + 3x\n\n3. **Division of a Polynomial by a Polynomial**: For Class 8, this typically involves factorising the dividend and then cancelling out any common factors with the divisor. If the dividend can be factorised into a product of expressions, and one of them is the divisor, then the other expression is the quotient.

Key facts to remember

1Factorisation is the process of writing an algebraic expression as a product of its factors.
2Always look for common factors first before trying other methods.
3The three standard algebraic identities are: (a + b)² = a² + 2ab + b², (a - b)² = a² - 2ab + b², and a² - b² = (a - b)(a + b).
4Factorisation by grouping terms is useful when there are no common factors among all terms.
5Division of algebraic expressions often involves factorising the dividend and then cancelling common factors with the divisor.
6When dividing monomials, divide coefficients and subtract powers of like variables.

Worked examples

Example 1

Factorise: 15x²y - 20xy²

IStep 1: Find the HCF of the numerical coefficients (15 and 20). The HCF of 15 and 20 is 5.

IIStep 2: Find the common literal factors with the lowest power. The common literal factors are 'x' (lowest power x¹) and 'y' (lowest power y¹). So, the common literal factor is xy.

IIIStep 3: The common factor of the entire expression is 5xy.

IVStep 4: Divide each term of the expression by the common factor 5xy:

V (15x²y) / (5xy) = 3x

VI (-20xy²) / (5xy) = -4y

VIIStep 5: Write the expression as the product of the common factor and the remaining expression.

VIII 15x²y - 20xy² = 5xy(3x - 4y)

Answer

5xy(3x - 4y)

Always look for the highest common factor for complete factorisation.

Example 2

Factorise: 49p² - 36q²

IStep 1: Observe the expression. It has two terms, both are perfect squares, and they are separated by a minus sign. This suggests using the identity a² - b² = (a - b)(a + b).

IIStep 2: Identify 'a' and 'b'.

III 49p² can be written as (7p)².

IV 36q² can be written as (6q)².

V So, a = 7p and b = 6q.

VIStep 3: Apply the identity a² - b² = (a - b)(a + b).

VII 49p² - 36q² = (7p)² - (6q)² = (7p - 6q)(7p + 6q)

Answer

(7p - 6q)(7p + 6q)

Recognising the form of the expression is crucial for applying the correct identity.

Example 3

Divide (x² + 8x + 16) by (x + 4).

IStep 1: Factorise the dividend (x² + 8x + 16).

II Observe that x² is a perfect square (x)², 16 is a perfect square (4)², and 8x is 2 * x * 4. This matches the identity a² + 2ab + b² = (a + b)².

III Here, a = x and b = 4.

IV So, x² + 8x + 16 = (x + 4)².

VStep 2: Rewrite the division problem using the factorised dividend.

VI (x² + 8x + 16) ÷ (x + 4) = (x + 4)² ÷ (x + 4)

VIIStep 3: Cancel the common factor (x + 4) from the numerator and the denominator.

VIII (x + 4)² / (x + 4) = (x + 4)(x + 4) / (x + 4) = x + 4

Answer

x + 4

Division of polynomials by polynomials in Class 8 often relies on factorising the dividend using identities or common factors.

Common mistakes

✗Incorrectly identifying the highest common factor (e.g., missing numerical factors or lowest powers of variables).
✗Making sign errors, especially when dealing with negative terms or applying identities like (a - b)².
✗Incorrectly applying algebraic identities (e.g., writing (a + b)² as a² + b²).
✗Cancelling terms that are not factors during division (e.g., cancelling 'x' from 'x + 4' and 'x + 2').
✗Not factorising completely (e.g., leaving a common factor inside the bracket).

Exam tips

★Memorise the three standard algebraic identities thoroughly, as they are frequently used in factorisation problems.
★Always check your factorisation by multiplying the factors back to see if you get the original expression.
★Practice identifying the correct factorisation method (common factors, grouping, or identity) for different types of expressions.
★Show all steps clearly in your solutions, especially in division problems, to avoid errors and gain full marks.

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