Introduction to Factorization

Factorization is the process of breaking down a number, algebraic expression, or polynomial into its simplest components, called factors. These factors, when multiplied together, give back the original number or expression.

Example in Numbers

For example, the number 12 can be factorized as:

12=2×2×3

Here, 2, 2, and 3 are the prime factors of 12.

Example in Algebra

In algebra, factorization is used to simplify expressions. For instance:

x²−5x+6=(x−2)(x−3)

Here, (x – 2) and (x – 3) are the factors of the quadratic expression.

Importance of Factorization

1. Simplifies Calculations – Helps in solving algebraic expressions easily.
2. Useful in Finding Roots – Factorization helps in solving quadratic and polynomial equations.
3. Applications in Real Life – Used in cryptography, physics, and engineering problems.

Factors of Natural Numbers

1. Definition of Factors

• A factor of a number is a natural number that divides it exactly without leaving a remainder.
• Example: Factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30 because they divide 30 completely.

2. Prime Factors

• Prime numbers are numbers that have only two factors: 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).
• Prime factors of a number are the factors that are prime.
• Example: Prime factors of 30 are 2, 3, and 5 because they are prime numbers.

3. Prime Factorization

• Writing a number as a product of its prime factors is called prime factorization.
• Examples:
  • 30 = 2 × 3 × 5
  • 70 = 2 × 5 × 7
  • 90 = 2 × 3 × 3 × 5

Factors of Algebraic Expressions

1. Definition of Factors in Algebraic Expressions

• In algebra, terms are formed as products of factors.
• Example: In the expression 5xy + 3x, the term 5xy is made of the factors 5, x, and y.
• Factors that cannot be broken down further are called irreducible factors (similar to prime factors in numbers).
• Example: 5 × x × y is the irreducible form of 5xy.

2. Concept of Irreducible Factors

• Irreducible factors cannot be split further into simpler factors.
• Example: 5xy = 5 × x × y (all three factors are irreducible).
• Incorrect irreducible form: 5 × (xy) (since xy can still be written as x × y).

3. Role of 1 in Factorization

• 1 is a factor of every term, but it is usually not written unless required.
• Example: 5xy = 1 × 5 × x × y, but we do not usually include 1.
• Similarly, 30 = 1 × 30, but 1 is not written unless necessary.

4. Factorizing Algebraic Expressions

• Expressions can be rewritten as a product of their irreducible factors.
• Example:
  • $3x(x+2)$ can be factorized as 3 × x × (x + 2).
  • $10x(x+2)(y+3)$ can be factorized as 10 × x × (x + 2) × (y + 3)