This module is about seeing the shape of an algebraic expression before you change it. You will collect like terms without touching the powers, expand brackets that contain negative terms, factorise with a common factor, factorise quadratics with whole-number factors, and simplify algebraic fractions while stating the values that are not allowed. Equations, formulas, graphs and sequences all rely on these moves.
Check the current Cambridge IGCSE Mathematics 0580 syllabus for the exact wording of each content point in your exam year. The habits here stay the same across versions. Our Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need signed-number arithmetic, the meaning of a power such as x², and how to multiply and divide simple numbers. The indices, roots and standard form module helps if powers still feel uncertain. If −3 × −4 still makes you hesitate, practise that first, because every lesson here depends on it.
An orienting example
Simplify (x² + 5x + 6) / (x² + 3x) and state the values of x that are not allowed.
Step 1, factorise the numerator: two numbers that multiply to 6 and add to 5 are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3).
Step 2, factorise the bottom: x² + 3x has common factor x, so it becomes x(x + 3).
Step 3, cancel the common factor (x + 3): this leaves (x + 2) / x.
Step 4, state exclusions: the original denominator x(x + 3) is zero when x = 0 or x = −3, so x ≠ 0 and x ≠ −3.
Check: put x = 1 into the original: (1 + 5 + 6) / (1 + 3) = 12/4 = 3. Put x = 1 into the answer: 3/1 = 3. Try x = 2 as well: 20/10 = 2 and 4/2 = 2. Both agree.
That one question used every lesson in the module. It shows why the order below builds from small moves to the full skill.
In which order should you study it?
- Collect terms without changing powers: the basic tidy-up, and the place where x² and x stop being interchangeable.
- Expand a product containing negative terms: brackets and signs, the source of most lost marks in algebra.
- Factorise using a common factor: expanding in reverse, and the first step of almost every factorising question.
- Factorise a quadratic with integer factors: the sum-and-product method, with the quadratic structure explorer to test your pair.
- Simplify an algebraic fraction with stated exclusions: pulls factorising together and adds the exclusion statement.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace. After this module, equations and formulas puts these moves to work.
Which traps catch most students here?
- Changing powers when collecting terms, such as writing 3x² + 2x² as 5x⁴.
- Losing a sign when a negative sits outside a bracket.
- Forgetting the 1 when a whole term is the common factor, so 5x² + 5x becomes 5x(x).
- Choosing the wrong signs when factorising a quadratic with a negative middle term.
- Cancelling across addition in a fraction, or leaving out the exclusions.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper before opening the answer. Write your working as you would in an exam, because method earns marks as well as the final expression. A quick substitution check, such as putting x = 1 into both the question and your answer, catches most slips in under a minute.
When something goes wrong, read the routing notes at the end of the practice set and return to the lesson it names. Fix the lesson, then try a fresh question a few days later. A simple written record of the error types is useful, and the mistake log and retest queue is one way to keep it.