Complete Guide For Class 8th Math Chapter 14: Factorization

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Factorization is a mathematical technique used to express a number or expression as a product of its factors. In Class 8, Chapter 14, students learn to factorize algebraic expressions, including monomials, binomials, and trinomials, using methods like common factors, grouping, and the difference of squares.

Factorization is an essential concept in mathematics, focusing on breaking down numbers, polynomials, or matrices into their constituent parts or factors. This chapter no. 14 of math introduces students to the methods and techniques used in factoring both numbers and algebraic expressions.

Introduction to Factorisation

Factorization involves decomposing an object into a product of simpler objects, or factors, which when multiplied together, recreate the original. For instance, the number 15 can be factored into primes as 3 × 5, and the polynomial x² − 4x² can be expressed as (x−2)(x+2).

The opposite process, known as expansion, involves multiplying factors to reform the original expression. Factorization aims to reduce objects to their “basic building blocks,” such as prime numbers for integers or irreducible polynomials for polynomial expressions.

Factors of Natural Numbers

To find the factors of a number, such as 30, we list all numbers that divide 30 evenly: 1, 2, 3, 5, 6, 10, 15, and 30. Among these, 2, 3, and 5 are the prime factors. Representing a number as a product of prime factors, like 30 = 2 × 3 × 5, is known as its prime factor form. This concept is extended to algebraic expressions.

Factors of Algebraic Expressions

In algebraic expressions, terms are formed as products of factors. For example, in the expression 5xy + 3x, the term 5xy consists of the factors 5, x, and y (i.e., 5xy = 5 × x × y). These factors, if irreducible, are considered ‘prime’ or ‘irreducible’ factors in algebra. For instance, the expression 3x(x + 2) can be factorised as 3, x, and (x + 2).

Method of Common Factors

When factorizing an algebraic expression, we express it as a product of its factors, which could be numbers, variables, or other algebraic expressions. For example, to factorize 2x + 4:

• Factor of 2x = 2 × x
• Factor of 4 = 2 × 2

So, 2x + 4 = 2(x + 2), with 2 and (x + 2) as factors.

Factorization by Regrouping Terms

In this method, terms in an algebraic expression are regrouped to facilitate taking common factors. For example, in 6xy – 4y + 6 – 9x, rearranging to 6xy – 4y – 9x + 6 allows us to group and factorize as 2y(3x – 2) – 3(3x – 2), leading to the factorised form (2y – 3)(3x – 2).

Factorization by Identity

Certain identities simplify the factorization process:

1. (a+b)² = a² + 2ab + b² 
2. (a−b)² = a² − 2ab + b²
3. (a+b)(a−b) = a² − b² 
4. (a+b)³ = a³ + 3a²b + 3ab² + b³ 
5. (a−b)³ = a³ − 3a²b + 3ab² − b³

These identities help in factorizing expressions like x² + 16x + 64 into (x + 8)².

Division of Algebraic Expressions

The division is the inverse of multiplication. For instance, dividing the monomial 6x³ by 2x involves canceling common factors, resulting in 3x². This principle extends to dividing polynomials by monomials or other polynomials.

Conclusion

Understanding factorization and the techniques involved is crucial for simplifying expressions and solving algebraic equations. This chapter no. 14 of Math Class 8th named Factorization equips students with the foundational knowledge to break down complex algebraic expressions into simpler components, a skill essential for higher-level mathematics.