Factorisation by Common Factor
Factorisation is the reverse of expansion. While expansion opens up brackets (e.g., 3(x + 2) = 3x + 6), factorisation puts them back (e.g., 3x + 6 = 3(x + 2)).
The simplest and most widely used method of factorisation is taking out the common factor. This means finding the factor that is common to all terms and writing the expression as a product.
This technique is the foundation of all factorisation methods. You must master it before moving to grouping, identities, or division methods.
What is Factorisation by Common Factor?
Definition: Factorisation by common factor means rewriting an algebraic expression as a product of its highest common factor (HCF) and the remaining expression.
ab + ac = a(b + c)
Where:
- a is the common factor of both terms
- b + c is the remaining expression after dividing each term by a
- The common factor can be a number, a variable, or both
Types of common factors:
- Numerical common factor: 6x + 12 = 6(x + 2)
- Variable common factor: x² + 3x = x(x + 3)
- Both numerical and variable: 4x² + 8x = 4x(x + 2)
Methods
Steps to factorise by taking out the common factor:
- Identify the HCF of all numerical coefficients.
- Identify the common variables — take the lowest power of each variable that appears in every term.
- The common factor = HCF of coefficients × common variables.
- Divide each term by the common factor to get the remaining expression.
- Write the result as: Common factor × (remaining expression).
How to find the HCF of coefficients:
- List the factors of each coefficient.
- Find the highest common factor.
- Example: Coefficients 12, 18, 24. Factors of 12 = {1,2,3,4,6,12}. Factors of 18 = {1,2,3,6,9,18}. Factors of 24 = {1,2,3,4,6,8,12,24}. HCF = 6.
How to find common variables:
- Take the lowest power of each variable present in ALL terms.
- Example: x³y², x²y³, x⁴y. Common: x² (lowest power of x) × y (lowest power of y) = x²y.
Solved Examples
Example 1: Example 1: Numerical common factor
Problem: Factorise: 6x + 18
Solution:
Steps:
- HCF of 6 and 18 = 6
- 6x ÷ 6 = x
- 18 ÷ 6 = 3
- 6x + 18 = 6(x + 3)
Check: 6(x + 3) = 6x + 18. Correct.
Answer: 6x + 18 = 6(x + 3)
Example 2: Example 2: Variable common factor
Problem: Factorise: x² + 5x
Solution:
Steps:
- Common variable: x (lowest power in both terms)
- x² ÷ x = x
- 5x ÷ x = 5
- x² + 5x = x(x + 5)
Check: x(x + 5) = x² + 5x. Correct.
Answer: x² + 5x = x(x + 5)
Example 3: Example 3: Both numerical and variable common factor
Problem: Factorise: 8x² + 12x
Solution:
Steps:
- HCF of 8 and 12 = 4
- Common variable: x (lowest power)
- Common factor = 4x
- 8x² ÷ 4x = 2x
- 12x ÷ 4x = 3
- 8x² + 12x = 4x(2x + 3)
Check: 4x(2x + 3) = 8x² + 12x. Correct.
Answer: 8x² + 12x = 4x(2x + 3)
Example 4: Example 4: Three terms
Problem: Factorise: 15x³ + 10x² + 5x
Solution:
Steps:
- HCF of 15, 10, 5 = 5
- Common variable: x (lowest power across all terms)
- Common factor = 5x
- 15x³ ÷ 5x = 3x²
- 10x² ÷ 5x = 2x
- 5x ÷ 5x = 1
15x³ + 10x² + 5x = 5x(3x² + 2x + 1)
Check: 5x(3x² + 2x + 1) = 15x³ + 10x² + 5x. Correct.
Answer: 5x(3x² + 2x + 1)
Example 5: Example 5: Two variables
Problem: Factorise: 6x²y + 9xy²
Solution:
Steps:
- HCF of 6 and 9 = 3
- Common variables: x (min power 1) and y (min power 1)
- Common factor = 3xy
- 6x²y ÷ 3xy = 2x
- 9xy² ÷ 3xy = 3y
6x²y + 9xy² = 3xy(2x + 3y)
Check: 3xy(2x + 3y) = 6x²y + 9xy². Correct.
Answer: 3xy(2x + 3y)
Example 6: Example 6: Subtraction with common factor
Problem: Factorise: 12a²b − 8ab²
Solution:
Steps:
- HCF of 12 and 8 = 4
- Common variables: a (min power 1) and b (min power 1)
- Common factor = 4ab
- 12a²b ÷ 4ab = 3a
- 8ab² ÷ 4ab = 2b
12a²b − 8ab² = 4ab(3a − 2b)
Check: 4ab(3a − 2b) = 12a²b − 8ab². Correct.
Answer: 4ab(3a − 2b)
Example 7: Example 7: Common factor is one of the terms
Problem: Factorise: 7x + 7
Solution:
Steps:
- HCF of 7 and 7 = 7
- 7x ÷ 7 = x
- 7 ÷ 7 = 1
7x + 7 = 7(x + 1)
Important: Do not forget the 1 inside the bracket. Writing 7(x) is wrong.
Answer: 7(x + 1)
Example 8: Example 8: Higher powers
Problem: Factorise: 20x⁴y³ − 15x³y⁴ + 10x²y²
Solution:
Steps:
- HCF of 20, 15, 10 = 5
- Common x: min(4, 3, 2) = x²
- Common y: min(3, 4, 2) = y²
- Common factor = 5x²y²
- 20x⁴y³ ÷ 5x²y² = 4x²y
- 15x³y⁴ ÷ 5x²y² = 3xy²
- 10x²y² ÷ 5x²y² = 2
= 5x²y²(4x²y − 3xy² + 2)
Check: Expand to verify it equals the original. Correct.
Answer: 5x²y²(4x²y − 3xy² + 2)
Example 9: Example 9: Negative common factor
Problem: Factorise: −6x² − 12x
Solution:
Steps:
- Common factor: −6x (taking out the negative)
- −6x² ÷ (−6x) = x
- −12x ÷ (−6x) = 2
−6x² − 12x = −6x(x + 2)
Alternatively: 6x(−x − 2) is also correct, but taking out the negative is neater.
Answer: −6x(x + 2)
Example 10: Example 10: Expression with constants only
Problem: Factorise: 18a + 24b + 30c
Solution:
Steps:
- HCF of 18, 24, 30 = 6
- 18a ÷ 6 = 3a
- 24b ÷ 6 = 4b
- 30c ÷ 6 = 5c
18a + 24b + 30c = 6(3a + 4b + 5c)
Check: 6(3a + 4b + 5c) = 18a + 24b + 30c. Correct.
Answer: 6(3a + 4b + 5c)
Real-World Applications
Real-world applications of factorisation:
- Simplifying algebraic expressions: Before solving equations, factorising can simplify complex expressions significantly.
- Solving equations: Factorised form helps find the roots of equations. If a × b = 0, then a = 0 or b = 0.
- Area calculations: Factorising helps in rewriting area expressions. If the area = 6x + 12 = 6(x + 2), the rectangle has dimensions 6 and (x + 2).
- Simplifying fractions: Factorising the numerator and denominator allows cancellation: (6x + 12)/(3x + 6) = 6(x + 2) / 3(x + 2) = 2.
- Engineering: Simplifying formulas in physics and engineering by factoring out common terms.
- Programming: Optimising calculations by factoring out common computations.
Key Points to Remember
- Factorisation is the reverse of expansion — writing an expression as a product of factors.
- The common factor method takes out the HCF of all terms.
- Common factor = HCF of coefficients × lowest power of each common variable.
- Always take out the highest common factor — do not leave further common factors inside the brackets.
- If a term is completely divided out, write 1 inside the bracket (not 0).
- To verify, expand the factorised expression — it should give back the original.
- The common factor can be a number, a variable, or both.
- This method is the first step in all factorisation problems — always check for common factors before trying other methods.
- Factorisation helps in simplifying fractions, solving equations, and understanding algebraic structure.
- Remember: ab + ac = a(b + c). This is the distributive property in reverse.
Practice Problems
- Factorise: 9x + 27
- Factorise: 4x² − 16x
- Factorise: 14a²b + 21ab²
- Factorise: 3x³ + 6x² + 9x
- Factorise: 25m²n − 15mn²
- Factorise: −8p² − 20p
- Factorise: 12x³y² + 18x²y³ − 6xy
- Simplify by factorising: (10x + 15) / (4x + 6)
Frequently Asked Questions
Q1. What is factorisation?
Factorisation is the process of writing an algebraic expression as a product of its factors. For example, 6x + 12 = 6(x + 2). It is the reverse of expanding brackets.
Q2. What is a common factor?
A common factor is a number, variable, or expression that divides every term of the algebraic expression exactly. In 8x + 12, the common factor is 4.
Q3. How do I find the common factor?
Find the HCF of all numerical coefficients, and take the lowest power of each variable that appears in every term. The product of these is the common factor.
Q4. What if the common factor equals one of the terms?
Then that term becomes 1 inside the bracket. For example, 5x + 5 = 5(x + 1). The 5 becomes 1, not 0. Never write 5(x + 0).
Q5. How do I verify my factorisation?
Expand the factorised form by multiplying the factor back into each term inside the brackets. If you get the original expression, your factorisation is correct.
Q6. Can I take out a negative common factor?
Yes. If all terms are negative, it is often cleaner to take out a negative common factor. For example, −6x − 12 = −6(x + 2).
Q7. What is the difference between factorisation and simplification?
Factorisation writes an expression as a product. Simplification reduces an expression to fewer terms. Factorisation is one technique used in simplification.
Q8. Why should I take out the HIGHEST common factor?
Taking the highest common factor ensures the expression inside the brackets cannot be factorised further. This gives the most simplified form.
Q9. Is factorisation by common factor always possible?
Only if all terms share a common factor other than 1. If the terms have no common factor (like 3x + 5y), this method does not apply. Other methods (grouping, identities) may work instead.
Q10. How is factorisation used in solving equations?
If an equation can be written as a(b) = 0, then either a = 0 or b = 0. Factorisation breaks expressions into simpler parts that can be set to zero individually.
