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1
What You'll Master
objectives
Know
What HCF (Highest Common Factor) means and how to find it
That factorising reverses the distributive law
Understand
Why factorising is the inverse of expanding
How to identify common factors in numerical and algebraic terms
Can Do
Fully factorise expressions with a numeric common factor
Fully factorise expressions with an algebraic common factor
Check answers by expanding back out
2
Words You Need
vocabulary
HCFHighest Common Factor — the largest factor that divides evenly into two or more numbers or terms.
FactoriseRewrite an expression as a product of factors. It is the reverse of expanding.
FactorA number or expression that divides exactly into another with no remainder.
Common FactorA factor that is shared by two or more numbers or terms. e.g. 3 is a common factor of 6 and 9.
Fully FactorisedAn expression where the HCF has been taken out completely — no common factor remains inside the brackets.
3
What is Factorising?
+5 XP
Factorising is the inverse of expanding. When you expand, you multiply out the brackets. When you factorise, you put the expression back into brackets by finding what is common to every term.
For example: $3(x + 2) = 3x + 6$ when you expand. So to factorise $3x + 6$, you reverse the process and get $3(x + 2)$.
Opposite operations
Expanding multiplies out. Factorising divides back in. They undo each other.
Check your answer
Always expand your factorised answer to verify you get back to the original expression.
4
Finding the HCF
+5 XP
Before you can factorise, you need to find the Highest Common Factor (HCF) of all the terms. There are two parts: the HCF of the numbers, and the HCF of any variable parts.
Step 1 — Numbers only: List the factors of each number and pick the largest one they share.
Find the HCF of 12 and 18:
Factors of 12: {1, 2, 3, 6, 4, 12}
Factors of 18: {1, 2, 3, 6, 9, 18}
Common factors: {1, 2, 3, 6}
HCF = 6 (the largest)
$\text{HCF}(12, 18) = 6$
Step 2 — Terms with variables: For variables, take the lowest power that appears in all terms.
Find the HCF of $6x$ and $9x^2$:
Number HCF: HCF(6, 9) = 3
Variable HCF: $x$ appears in both; lowest power is $x^1$
Overall HCF = $3x$
Rule for variables
HCF uses the lowest power. HCF of $x^3$ and $x^2$ is $x^2$, not $x^3$.
Variable must be in ALL terms
If one term has no $x$, then $x$ cannot be part of the HCF.
5
Factorising Algebraic Expressions
+5 XP
Follow this three-step method every time you factorise:
Find the HCF of all terms.
Divide each term by the HCF.
Write: HCF × (remaining terms in brackets).
Try It Now: Factorise $12x + 16$. What is the HCF? What goes inside the brackets?
Answer: HCF = 4. So $12x + 16 = 4(3x + 4)$. Check: $4 \times 3x = 12x$ and $4 \times 4 = 16$. ✓
Watch Me Solve It · Worked example
Watch Me Solve It · Fully factorise $12x^2 + 8x$
+15 XP per step
Q
PROBLEM
Fully factorise $12x^2 + 8x$. Show all working and verify your answer.
1
Find the HCF of the numbers
HCF(12, 8) = 4
Factors of 12: {1, 2, 3, 4, 6, 12}. Factors of 8: {1, 2, 4, 8}. Largest common factor is 4.
2
Find the HCF of the variable parts
HCF($x^2$, $x$) = $x$
$x^2$ has power 2; $x$ has power 1. Take the lowest power — that is $x^1 = x$. Since $x$ appears in both terms, it is part of the HCF.
3
Combine: overall HCF = $4x$. Divide each term.
$12x^2 \div 4x = 3x \qquad 8x \div 4x = 2$
Divide the coefficient by 4 and reduce the power of $x$ by 1 (i.e. divide by $x$) for each term.
4
Write the answer and verify by expanding
$12x^2 + 8x = 4x(3x + 2)$
Check: $4x \times 3x = 12x^2$ and $4x \times 2 = 8x$. Adding: $12x^2 + 8x$ ✓. The answer is fully factorised because $3x$ and $2$ share no common factor.
Nice work — XP earned
Answer$12x^2 + 8x = 4x(3x + 2)$
6
Common Pitfalls
heads-up
Not taking out the FULL common factor
Writing $2(6x^2 + 4x)$ instead of $4x(3x + 2)$ for $12x^2 + 8x$. The expression is not fully factorised if a common factor still remains inside the bracket.
Fix: always check — can any number or variable be divided out of every term inside the bracket? If yes, you have not finished yet.
Forgetting to include the variable in the HCF
For $6x^2 + 9x$, writing HCF = 3 and getting $3(2x^2 + 3x)$ — but the $x$ is still common! The correct HCF is $3x$, giving $3x(2x + 3)$.
Fix: after finding the number HCF, always ask "does every term share at least one copy of each variable?" If so, include it.
Forgetting to check by expanding
Many errors go undetected because students skip the verification step. Expanding your answer takes 10 seconds and catches mistakes.
Fix: always expand your factorised answer. If you get back the original expression, you are done. If not, find the error.
Quick Check · 5 questions
1
What is the HCF of $8x$ and $12$?
+10 XP
2
Factorise $5x + 10$.
+10 XP
3
Factorise $x^2 + 7x$.
+10 XP
4
Factorise $6a^2 + 9a$ completely.
+10 XP
5
Which expression is fully factorised?
+10 XP
ApplyEasy2 MARKS
Q1. Factorise $7x + 14$.
Answer in your workbook.
ApplyMedium3 MARKS
Q2. Factorise $10y^2 - 15y$ completely.
Answer in your workbook.
ApplyHard4 MARKS
Q3. Factorise $3ab + 6a^2b - 9ab^2$ completely.
Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins
Extension Problems
Factorise $12x^2y + 8xy^2 - 4xy$ completely. (Hint: check all three parts — number, x, and y!)
R
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