Factorisation

Detailed Notes on Factorisation

1. Introduction to Factorisation

Factorisation is the process of breaking down a number or an algebraic expression into its constituent factors, which are the numbers or expressions that multiply together to produce the original number or expression.

1.1 Factors of Natural Numbers

• A factor is a number that divides another number without leaving a remainder. For example, the factors of 30 include 1, 2, 3, 5, 6, 10, 15, and 30.
• Prime factors are factors that are prime numbers; for example, the prime factorization of 30 is 2 × 3 × 5.

2. Factors of Algebraic Expressions

In a similar way, algebraic expressions can have their terms broken down into factors:

• Example: For the term 5xy, the factors are 5, x, and y.
• Here, terms refer to individual parts of an expression that are added or subtracted.

3. What is Factorisation?

While factorisation generally refers to breaking down numbers, in algebra, it means writing an expression as a product of its algebraic factors.

3.1 Common Factors

The common factor method involves searching for the greatest common factor among the terms of an expression:

• Example: For 2x + 4, each term can be rewritten as:
  • 2x = 2 × x
  • 4 = 2 × 2
  • Thus, 2x + 4 can be factored into 2(x + 2).

4. Regrouping Terms

When there’s no common factor among all terms, a method called regrouping can be employed:

• This involves rearranging terms to find patterns that allow extraction of common factors.
• Example: For 2xy + 2y + 3x + 3, we can regroup as (2xy + 2y) + (3x + 3), yielding the common factor 2y and 3, subsequently allowing the full expression to be factored as (x + 1)(2y + 3).

5. Factorisation Using Identities

Identities are fixed mathematical truths that can be applied for factorisation:

• Common identities include: 1. (a + b)² = a² + 2ab + b²
  2. (a - b)² = a² - 2ab + b²
  3. (a + b)(a - b) = a² - b²
• Recognizing these forms can streamline the factorisation process.
• For example, to factor x² + 8x + 16, you can recognize it as (x + 4)² since it matches a² + 2ab + b² with a = x, b = 4.

6. Factors in Terms of One Variable

When dealing with quadratic expressions like x² + px + q, it's helpful to find two numbers that multiply to q (the constant term) and add to p (the coefficient of x).

• Example: For x² + 5x + 6, the factors are (x + 2)(x + 3) because 2×3 = 6 and 2 + 3 = 5.

7. Division of Algebraic Expressions

Understanding division is crucial: it inversely relates to multiplication.

7.1 Monomial Division

• When dividing monomials, simplify by canceling common factors, just like dividing numbers. Example:
  • 24x² ÷ 12x = 2x (dividing coefficients and then reducing the variable powers).

7.2 Polynomial by Monomial

• Divide each term in a polynomial by the monomial or factor out the common terms, simplifying each individual part before reconstruction.

8. General Practices in Factorisation

• Look for common factors
• Regroup terms strategically.
• Use identities as needed.
• Check your work for correctness through expansion of the factors.

Conclusion

Mastering factorisation is foundational in algebra, offering insight into the structure of numbers and expressions, enabling simplification and easier calculation in more complex scenarios.

1. Factorisation transforms expressions into products of factors.
2. Common Factors are essential for simplifying terms in polynomials.
3. Regrouping may help identify commonality for factorisation.
4. Identities provide formulaic methods for factorisation.
5. Division maintains the relationship of expression simplification.
6. A systematic approach aids in keeping track of factors and operations.
7. Focus on finding the greatest common factor to simplify effectively.
8. Ensure accuracy through verification steps post-factorisation.