Factorising, Common Factor
Expanding runs one way; factorising runs the other. Find the highest common factor of every term, take it outside a bracket, and check by expanding back.
Printable Worksheets
Print or save as PDF, or build a custom worksheet from any module's questions.
Before diving into the practice questions, take a moment to think about what you already know about this topic.
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
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 $3(x + 2)$.
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.
Start by listing the factors of a single term. A factor must divide the term exactly: the positive monomial factors of $6x^2$ are $1$, $2$, $3$, $6$, $x$, $2x$, $3x$, $6x$, $x^2$, $2x^2$, $3x^2$ and $6x^2$. Listing coefficient factors and variable factors systematically helps you see what different terms share.
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)
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 = $3x$
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).
Watch Me Solve It · Worked example
-
1Find the HCF of the numbersHCF(12, 8) = 4Factors of 12: {1, 2, 3, 4, 6, 12}. Factors of 8: {1, 2, 4, 8}. Largest common factor is 4.
-
2Find the HCF of the variable partsHCF($x^2$, $x$) = $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.
-
3Combine: 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.
-
4Write 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.
Quick Check · 5 questions
Q1. Factorise $7x + 14$.
Q2. Factorise $10y^2 - 15y$ completely.
Q3. Factorise $3ab + 6a^2b - 9ab^2$ completely.
Extension Problems
Factorise $12x^2y + 8xy^2 - 4xy$ completely. (Hint: check all three parts, number, x, and y!)
Key Concept
Review the main ideas from this lesson.
Formulas
Key formulas and rules.
Watch Out
Common mistakes to avoid.
Check
Always verify your answers.
Practice
Keep practicing to master.
Next
Build on these skills.
Interactive: Algebra Machine
Substitute numbers into algebraic expressions and see them evaluate step by step.
Your Badges
0 of 6Mark lesson as complete
Tick when you've finished Learn, Practice and the Stretch. Earns +90 XP and +25 coins.