Factorise by taking out a common factor
Find a shared numerical and variable factor · Rewrite a sum as a product · Verify by expanding back
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- GCF
- Algebraic powers and expansion
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Find a shared numerical and variable factor
- Rewrite a sum as a product
- Verify by expanding back
Reverse distribution
To factorise, identify a factor shared by every term and divide each term by it inside parentheses. In 8x²+12x, the common factor 4x leaves 2x+3, so the expression is 4x(2x+3). The equality also holds at x = 0 because no original denominator is introduced.
Keep a complete bracket
A term equal to the common factor leaves 1, not an empty place: 6x+6 = 6(x+1). Negative common factors are possible if every sign is changed consistently. Expand the final product to check every original term. Core requires common-factor factorisation; other patterns are Extended lessons.
Sources for this lesson
Pause and explain
What is a factorised form of 6x+6?
Worked example
Factorise 15a²−10a.
Show the worked solution
- The shared numerical factor is 5 and both terms contain a.
- Divide each term by 5a to get 3a−2.
Answer 5a(3a−2).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Factorise 12x+18.
Give me a hint
Find the greatest shared integer factor.
Compare my reasoning
- GCF is 6.
- 12x+18 = 6(2x+3).
6(2x+3).
Look for these in your work
- Keep both terms
- Expand to verify
Factorise −8y²+12y.
Give me a hint
A common factor is 4y.
Compare my reasoning
- Divide terms by 4y to obtain −2y+3.
- Expand 4y(−2y+3) to recover the original.
4y(−2y+3).
Look for these in your work
- Retain signed coefficients
- Check the product
An expression for area is 6x²+9x with x > 0. Factorise it to suggest two side lengths and verify at x = 2.
Give me a hint
Extract 3x.
Compare my reasoning
- 6x²+9x = 3x(2x+3), both positive when x > 0.
- At x = 2 the factors are 6 and 7, whose product 42 matches 24+18.
Sides 3x and 2x+3; area 42 at x = 2.
Look for these in your work
- Use a product interpretation
- Check positivity and a numerical value
Common mistakes
- An extracted full term leaves a 1 inside.
- Check the signs by expansion.
What can you explain now?
What is a factorised form of 6x+6?
Compare with the explanation
6(x+1).
Both terms share 6; the constant term becomes 1 inside.
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Make it stick
What is a factorised form of 6x+6?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.