Simplify, expand and factorise algebraic expressions
Collect like terms, expand brackets and reverse multiplication through factorisation, with Core and Extended scope made clear.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which terms can combine in 4x + 3x² − 2x?
What you will learn
- Collect like terms and multiply algebraic terms
- Expand brackets and factorise common factors
- Use Extended patterns including quadratics and difference of squares
Simplify without mixing unlike terms
Only terms with identical variable parts combine: 5a + 2a − 3a = 4a, but a and a² stay separate. When multiplying, multiply coefficients and add powers with the same base.
Expand a bracket by multiplying every term inside it. Two brackets require every term in the first to multiply every term in the second.
Factorisation reverses expansion
Take out the highest common factor first. For quadratics, look for factor pairs whose product and sum match the constant and x coefficient.
Extended work also includes grouping, products of more than two brackets, difference of squares, perfect squares, quadratics and cubics with a common factor. Expand back to check.
Pause and explain
Expand 4(2y − 3).
Worked example
Expand (2x − 3)(x + 4), then factorise x² + 5x + 6.
Show the worked solution
- Multiply all four pairs: 2x² + 8x − 3x − 12.
- Collect like terms: 2x² + 5x − 12.
- Numbers 2 and 3 multiply to 6 and add to 5.
Answer 2x² + 5x − 12; (x + 2)(x + 3).
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.
Expand 3(2x − 5) + 2(x + 4), then factorise 6x² − 9x.
Give me a hint
Expand both brackets before collecting; then identify the common factor.
Compare my reasoning
- The brackets give 6x − 15 and 2x + 8.
- Together this is 8x − 7.
- 6x² − 9x = 3x(2x − 3).
8x − 7; 3x(2x − 3).
Look for these in your work
- Both brackets expanded
- Like terms collected
- Factorisation multiplies back correctly
Factorise x² − 9 and x² + 7x + 12; expand (x + 2)(x − 5).
Give me a hint
Use difference of squares, then check factor pairs by sum and product.
Compare my reasoning
- x² − 9 = (x − 3)(x + 3).
- x² + 7x + 12 = (x + 3)(x + 4).
- (x + 2)(x − 5) = x² − 3x − 10.
(x − 3)(x + 3); (x + 3)(x + 4); x² − 3x − 10.
Look for these in your work
- Difference of squares used
- Quadratic pair checked
- Four products included
A rectangle has sides x + 3 and x − 2 metres. Find its area expression. Then factorise x² + 4x + 3.
Give me a hint
Multiply dimensions; for the quadratic find factors of 3 that add to 4.
Compare my reasoning
- Area = (x + 3)(x − 2).
- Expanding gives x² + x − 6 m².
- x² + 4x + 3 = (x + 1)(x + 3).
Area x² + x − 6 m²; factors (x + 1)(x + 3).
Look for these in your work
- Dimensions multiplied
- Terms collected
- Factors satisfy sum and product
Common mistakes
- Combining different powers
- Expanding only the first term
- Losing a negative sign
- Stopping before factorisation is complete
What can you explain now?
Factorise 8x² − 12x fully.
Compare with the explanation
4x(2x − 3)
Take out the highest common factor 4x.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How can reversing expansion check that a factorisation is correct?
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.