Rewrite expressions by taking out common factors

Introduction

• Essential knowledge for factorising includes understanding factors and highest common factors (HCF), as well as having a solid grasp of divisibility rules.
• An expression rewritten by extracting common factors is considered factorised.
• Factorising is the reverse process of expanding brackets.
• A factorised expression is equivalent to the original expression, making them identities. The identity symbol (≡) is used to link the original expression and its equivalent factorised expression.
• Proper understanding and use of algebraic notation are necessary to accurately complete the factorising process.

Finding the Highest Common Factors (HCF) in an Expression

When the expression includes a constant, the highest common factor (HCF) will be a number:

• Find the HCF of the coefficients and the constant, which is the greatest factor common to all the numbers in the expression.
• The HCF for the expression is a number.

When the expression has no constants:

• Find the HCF of the coefficients, the greatest factor common to all the coefficients in the expression.
• Identify common variable factors in each term, which can be a single variable or a combination of variables.
• The HCF of the expression is the product of the number and variable HCFs.

Example

Example: Find the HCF of this expression.
Sol:

• The expression has no constants. The HCF of the coefficients is the greatest factor that is common to all the numbers in the expression. The coefficients are 15, 6 and 21. The factors of 15 are 1, 3, 5 and 15. The factors of 6 are 1, 2, 3 and 6. The factors of 21 are 1, 3, 7 and 21. The HCF of 15, 6 and 21 is 3 
  
• The common variable in 15𝒂² + 6𝒂 + 21𝒂𝒃 is 𝒂. The variable HCF is 𝒂
  
• The HCF of the expression is the number HCF (3) multiplied by the variable HCF (𝒂). 3 × 𝒂 = 3𝒂. The HCF of the expression is 3𝒂

Factorizing Simple Expressions

• When the expression includes a constant, the highest common factor (HCF) is a number.
• When the expression does not include a constant, the HCF can be a number, a variable, or a combination of both.
• To factorise an expression:
  • Find the HCF of the numbers in the expression.
  • Find the HCF of the variables in the expression.
  • Multiply the number and variable HCFs, and write this term in front of the bracket.
  • Determine the terms inside the bracket by completing the factor pair for each of the original terms in the expression.
• The identity symbol (≡) is used to link the original expression and its equivalent factorised expression.

Example

Example: Factorise 12𝒂𝒃 - 3𝒂
Sol:

• The highest common factor of 12 and 3 is 3. The HCF of the variable terms 𝒂𝒃 and 𝒂 is 𝒂. The HCF of 12𝒂𝒃 and 3𝒂 is the product of 3 and 𝒂 (3𝒂).
  
• 3𝒂 is the HCF of 12𝒂𝒃 and 3𝒂. 3 is the highest common factor of 12 and 3. 𝒂 is the highest common factor of 𝒂𝒃 and 𝒂. 3𝒂 is written in front of the bracket. The first term in the bracket is 4𝒃 because 3𝒂 × 4𝒃 = 12𝒂𝒃. The second term in the bracket is -1 because 3𝒂 × -1 = -3. 12𝒂𝒃-3𝒂 is factorised as the equivalent expression 3𝒂(4𝒃-1).