Common Factors

Learn how to factorise algebraic expressions using common factors with simple student-friendly notes, steps, examples, and practice.

1. Introduction

Factorisation means breaking an expression into simpler expressions called factors. Just like numbers can be written as products (e.g., 12 = 3 × 4), algebraic expressions can also be split into factors.

In this method, we look for a common factor in every term and take it out.

2. What Is a Factor?

A factor is a number, variable, or expression that divides another expression exactly.

  • Numerical factors: 2, 3, 5, etc.
  • Variable factors: \(x\), \(y\), \(x^2\), etc.
  • Algebraic factors: \((x+2)\), \((3y-5)\)

3. Finding Common Factors

To factorise an expression using the common factor method, find what is common in all terms.

3.1. 1. Common Numerical Factors

These come from the numbers in each term.

Example: In \(6x\) and \(9x^2\), the common numerical factor is \(3\).

3.2. 2. Common Variable Factors

Choose the smallest power of each variable appearing in all terms.

Example: In \(5x^3\) and \(10x^2\), the common variable factor is \(x^2\).

3.3. 3. Common Algebraic Factors

Sometimes entire brackets repeat.

Example: In \(3(x+2)\) and \(7(x+2)\), the common factor is \((x+2)\).

4. Steps to Factorise Using Common Factors

  1. Identify common numerical factors.
  2. Identify common variable factors.
  3. Take the common factor out.
  4. Write the expression as:
    Common factor × Remaining expression

4.1. Example Walkthrough

Factorise: \(12x^2y\) and \(18xy^2\)

Step 1: Common number = \(6\)

Step 2: Common variables = \(xy\)

So common factor = \(6xy\)

Now divide each term:

  • \(12x^2y ÷ 6xy = 2x\)
  • \(18xy^2 ÷ 6xy = 3y\)

Final answer:
\(6xy(2x + 3y)\)

5. More Examples

1. Factorise: \(4x + 20\)

Common factor = \(4\)

Answer: \(4(x + 5)\)

2. Factorise: \(15a^2 - 10a\)

Common factor = \(5a\)

Answer: \(5a(3a - 2)\)

3. Factorise: \(9xy + 6x^2y\)

Common factor = \(3xy\)

Answer: \(3xy(3 + 2x)\)

6. Common Mistakes

  • Forgetting to take the highest common factor.
  • Taking out a factor that is not common to all terms.
  • Dropping variables accidentally.
  • Incorrect dividing when finding the remaining expression.

7. Quick Practice

Factorise:

  1. \(8x + 12\)
  2. \(21ab - 14ac\)
  3. \(5x^2y + 10xy^2\)
  4. \(18p^2q - 9pq^2\)

8. Summary

  • Factorisation by common factor is the simplest method.
  • Find common numerical, variable, and algebraic factors.
  • Final form is always: \( \text{common factor} \times \text{remaining expression} \).