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Detailed notes on Algebra for Cambridge Lower Secondary Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.
The things students keep getting wrong in this sub-topic, answered.
Revise how to expand single and double brackets, and how to factorise expressions by spotting common factors and simple quadratic patterns — building confidence with both directions so your algebra stays tidy heading into your Checkpoint.
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
Multiply every term inside the bracket by the factor outside — none gets left behind.
Expanding a bracket means rewriting it without the brackets, by multiplying every term inside by the factor outside. The rule that powers this is the distributive law: .
A few quick examples set the pattern:
Multiply every term in the first bracket by every term in the second — four products in total.
When two brackets sit side by side, like , you have to multiply every term in the first by every term in the second. The friendly memory trick is FOIL:
Take out the biggest factor that divides every term — number and letters.
Factorising is the reverse of expanding. Instead of removing brackets, you put them back by spotting what each term has in common.
For an expression like , both terms share a factor of . Take it out: . The inside of the bracket is what is left after dividing each original term by .
factorises to , where and .
Expand your factorised answer back out — it should match what you started with.
Whether you've expanded or factorised, the single most useful check is to do the opposite operation and see whether you land back where you began.
If you expanded and got , factorise it back: two numbers that multiply to and add to — that's and , giving . The loop closes.
Expanding and factorising are the engines of all later algebra.
Confident expansion and factorising open the door to:
Sources: Cambridge Lower Secondary Mathematics curriculum framework. Last reviewed 2026-05-19.
Step-by-step solutions for expansion and factorising, written exactly the way a tutor would explain them at the board.
Question
Expand .
Step-by-step solution
Step 1
Multiply each term inside the bracket by the outside factor.
Step 2
Write the two products as one expression.
Answer
Question
Factorise completely.
Step-by-step solution
Step 1
Find the largest number that divides both and . It is .
Question
Expand and simplify .
Step-by-step solution
Question
Factorise completely.
Step-by-step solution
Question
Factorise .
The important words for expansion and factorising and what they mean — learn these so you can explain your thinking clearly.
To remove brackets by multiplying each inside term by the outside factor. So .
To rewrite an expression as a product of factors — the reverse of expanding. So .
A factor (number or letter) that divides exactly into every term of an expression. The largest one is taken out when factorising.
The round symbols and that group terms together. Anything multiplying a bracket applies to every term inside it.
A memory aid for expanding double brackets: First, Outside, Inside, Last — the four pairings of terms whose products are added together.
An expression whose highest power is , for example .
The number in front of a variable. In the coefficient is .
A term in an expression that has no variable — just a number. In the constant term is .
The rule that lets you expand a bracket: . Multiplication distributes over addition.
Terms with the same variable parts and the same powers. After expanding, only like terms can be combined.
The result of multiplying terms together. In factorising, you write an expression as a product of factors.
The slip-ups students most often make with expansion and factorising — and simple ways to avoid them.
Why it happens
Attention drops once the first multiplication is done.
How to avoid it
Picture an arrow from the outside factor to each inside term in turn — every term gets multiplied.
Why it happens
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After expanding, collect like terms if the bigger expression has any. For example . Two warnings worth remembering: a minus outside a bracket flips every inside sign, and a missing operation between the factor and the bracket is always multiplication.
Add the four products together: . The two middle terms always collect into one.
Watch the signs carefully. For , the four products are , , and , giving . The "Last" pair is the constant term and inherits the signs of both bracketed numbers.
When letters are involved too, take the largest number that divides every coefficient and the lowest power of each letter that appears in every term.
After factorising, check by expanding your bracket back out. If it returns to the original expression, you've factorised completely. If you can still see another common factor inside the bracket, you have not gone far enough.
A simple quadratic is an expression of the form — three terms, with as the highest power and no number in front of the . These factorise into a pair of brackets , where two clever numbers and must satisfy:
So to factorise , look for two numbers that multiply to and add to . Try the pairs: and (no, adds to ), and (no, adds to ), and (yes, adds to ). So .
Signs matter. If is positive, both and have the same sign as . If is negative, and have , and the bigger one carries the sign of .
As always, check by expanding your factorised answer.
If you factorised and got , expand: , . Returns to the original — the answer is good.
One last habit. After factorising, scan the inside of the bracket for any further common factor. If you can spot one, factorise again. The aim is fully factorised, not just partly so.
If a later topic feels slow, it is often a missed sign in an expansion or an incomplete factorisation underneath. Come back to this guide whenever you need a steady reset — accuracy beats speed every time.
Step 2
Take outside a bracket and divide each original term by to fill the bracket.
Step 3
Write the factorised form and check by expanding.
Answer
Step 1
First terms.
Step 2
Outside and Inside terms.
Step 3
Last terms.
Step 4
Collect the two middle terms.
Step 5
Write the tidy answer.
Answer
Step 1
Find the largest number that divides both coefficients: .
Step 2
Both terms contain at least one , so the lowest power of in common is .
Step 3
The common factor is . Divide each term by .
Step 4
Write the factorised form.
Answer
Step-by-step solution
Step 1
Look for two numbers whose product is and whose sum is .
Step 2
Try pairs that multiply to : , , , .
Step 3
Check which pair sums to .
Step 4
Write the factorised form using those two numbers.
Step 5
Check by expanding: ✓
Answer
Why it happens
Only the First and Last products of FOIL are remembered; Outside and Inside go missing.
How to avoid it
Always write all four products before tidying: .
The negative is only applied to the first inside term.
How to avoid it
A negative outside flips every inside sign: .
Why it happens
The number factor is taken out but the shared is missed.
How to avoid it
Check the inside bracket for further common factors. Here still divides both terms, so the full factorisation is .
Why it happens
The two numbers are chosen but their signs are swapped.
How to avoid it
If the constant is negative, the two numbers have opposite signs; the larger one matches the sign of the middle term. Here and give a sum of , so the answer is .