Expanding and Factorising Algebra Practice Questions
Build confidence with the main expanding and factorising techniques used in secondary algebra. Each section contains 10 questions with instant checking, helpful hints and complete working.
- Expanding: single brackets, double brackets and collecting like terms
- Common factors: factorise numerical and algebraic expressions using the highest common factor
- Quadratics: factorise monic and non-monic quadratic expressions
- Special products: recognise and factorise a difference of two squares
- Mixed practice: decide whether to expand, simplify or factorise
- 8 focused levels
- 80 practice questions
- Instant answer checks
- Hints and full working
Spaces do not affect marking. Common equivalent factor orders and x² or x^2 notation are accepted. Your answers and checked results are saved automatically in this browser.
Level 1
Expanding Single Brackets
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Level 2
Expanding and Collecting Like Terms
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Level 3
Expanding Double Brackets
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Level 4
Factorising by a Common Factor
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Level 5
Factorising Monic Quadratics
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Level 6
Factorising Harder Quadratics
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Level 7
Difference of Two Squares
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Level 8
Mixed Expanding and Factorising Challenge
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Expanding and Factorising Algebraic Expressions
Expanding removes brackets by multiplying, while factorising rewrites an expression as a product of factors. These opposite processes help students simplify algebra, solve quadratic equations and understand the equations of parabolas.
- expand single and double brackets;
- collect like terms correctly;
- factorise using a highest common factor;
- factorise monic and non-monic quadratics;
- recognise difference-of-squares patterns;
- check factorised answers by expanding.
Quick Summary
| Skill | What it means | Example |
|---|---|---|
| Expanding | Remove brackets using multiplication. | 3(x + 4) = 3x + 12 |
| Factorising | Rewrite an expression as a product. | 3x + 12 = 3(x + 4) |
| Double brackets | Multiply every term in one binomial by every term in the other. | (x + 2)(x + 3) = x² + 5x + 6 |
| Difference of squares | Use the pattern a² − b² = (a − b)(a + b). | x² − 25 = (x − 5)(x + 5) |
How Expansion and Factorisation Fit the NSW Syllabus
Expansion and factorisation sit within the NSW Mathematics K–10 focus areas of Algebra and equations and Linear and non-linear relationships. Students generally develop foundational algebraic techniques during Stage 4 and extend them into quadratic expressions, equations and parabolas during Stage 5.
Stage 4: Years 7–8
Students commonly strengthen the distributive law, simplifying expressions, collecting like terms and taking out common factors. Review the Year 7 NSW Maths guide and Year 8 NSW Maths guide.
Stage 5: Years 9–10
Students may progress to binomial products, quadratic expressions, special factorisation patterns and applications involving equations and graphs. See the Year 9 guide and Year 10 guide.
1. Expanding Single Brackets
To expand a single bracket, multiply the term outside the bracket by every term inside. This is the distributive law:
The 8 multiplies both x and 5.
Expanding with a negative multiplier
A negative multiplier changes the signs of the terms it multiplies:
2. Expanding and Collecting Like Terms
After expanding, combine terms that have exactly the same pronumeral part and power. For example, 4x and 6x are like terms, but x and x² are not.
Only the coefficients of like terms are added or subtracted. Constants combine with constants, x-terms with x-terms, and x²-terms with x²-terms.
3. Expanding Double Brackets
Multiply every term in the first bracket by every term in the second bracket. FOIL—First, Outer, Inner, Last—is a useful memory aid for two binomials, but the distributive law is the underlying method.
4. Factorising by a Common Factor
Factorising is the reverse of expanding. Find the highest common factor shared by every term, place it outside the bracket and divide each term by it.
Always look for a common factor before trying another method. An expression is not fully factorised while its terms still share a common factor.
5. Factorising Monic Quadratics
A monic quadratic has a coefficient of 1 in front of x². Its general form is:
Find two numbers that:
- multiply to give c; and
- add to give b.
The numbers 2 and 3 multiply to 6 and add to 5.
6. Factorising Non-Monic Quadratics
A non-monic quadratic has a leading coefficient other than 1:
One reliable method is to multiply a and c, find two numbers that multiply to ac and add to b, split the middle term, then factorise by grouping.
Here, a × c = 2 × 3 = 6. The numbers 6 and 1 multiply to 6 and add to 7.
Check the result by expanding:
7. Difference of Two Squares
A difference of two squares contains two perfect-square terms separated by subtraction:
8. Perfect-Square Trinomials
A useful related pattern occurs when the first and last terms are squares and the middle term is twice their product:
The first term is x², the last term is 5², and the middle term is 2 × x × 5 = 10x.
How to Choose the Correct Method
Look for brackets
If the instruction says expand, multiply to remove the brackets and collect like terms.
Look for a common factor first
Before factorising a quadratic or using a special pattern, check whether every term shares a numerical or algebraic factor.
Identify the expression type
Decide whether it is a monic quadratic, non-monic quadratic, difference of squares or perfect-square trinomial.
Verify the answer
Expand the factorised expression. It should reproduce the original expression exactly.
Why Factorising Matters for Equations and Parabolas
Factorising is not only a simplification skill. It is used to solve quadratic equations through the zero-product property:
In a parabola equation such as
Common Mistakes to Avoid
- Multiplying only the first term inside a bracket.
- Losing a negative sign when distributing a negative multiplier.
- Combining unlike terms such as x and x².
- Forgetting one of the four products when expanding double brackets.
- Not taking out the highest common factor first.
- Using numbers that multiply correctly but do not add to the middle coefficient.
- Stopping before an expression is fully factorised.
- Using difference of squares when the terms are being added.
- Failing to verify a factorised answer by expanding it.
Revision Checklist
- I can apply the distributive law to every term inside a bracket.
- I can collect like terms after expanding.
- I can expand two binomials systematically.
- I look for a highest common factor before other methods.
- I can factorise monic and non-monic quadratics.
- I recognise difference-of-squares and perfect-square patterns.
- I can check factorisation by expanding.
Frequently Asked Questions
What is the difference between expanding and factorising?
Expanding removes brackets by multiplication. Factorising reverses the process by rewriting an expression as a product of factors.
Should I always take out the common factor first?
Yes. Checking for a highest common factor first usually makes the remaining expression simpler and helps ensure the final answer is fully factorised.
Does the order of factors matter?
No. Multiplication is commutative, so (x + 2)(x + 3) and (x + 3)(x + 2) are equivalent.
Can every quadratic expression be factorised using integers?
No. Some quadratics do not have integer factors. Depending on the question, students may need the quadratic formula, completing the square or another method.
Why is factorising useful for parabolas?
Factorised form can reveal the x-intercepts of a parabola. It also helps solve quadratic equations associated with the graph.
Related Aussie Math Tutor NSW Resources
Trusted external references
Check the official NSW Mathematics K–10 syllabus overview, the official NSW Mathematics glossary, and the NSW Department of Education’s Years 7–10 units and assessments.
For additional worked explanations, students can review factoring polynomials at OpenStax, quadratic factorisation at Khan Academy, and expanding algebra at Maths Is Fun.
Still Finding Expansion or Factorisation Difficult?
Students often know one method but become unsure when signs change, brackets are doubled or the quadratic is non-monic. A free assessment can identify the exact algebra skills that need strengthening.
View Free Assessment TimesFrequently Asked Questions about Expanding and Factorising algebraic expressions
What expansion and factorisation methods do NSW Year 9 and Year 10 students need to know?
Year 9 students usually need to understand how to expand brackets and factorise simple algebraic expressions.
For expansion, students learn how to multiply a term outside the bracket with every term inside the bracket. For example:
3(x + 4) = 3x + 12
For factorisation, students learn how to find the common factor and place it outside the bracket. For example:
6x + 12 = 6(x + 2)
Some schools may also introduce more advanced algebra, such as expanding double brackets and factorising simple quadratic expressions. However, many students are expected to become more confident with quadratic expansion and factorisation in Year 10.
Why is factorising quadratics so important in the NESA Stage 5 syllabus?
Factorising quadratics is important because quadratic equations are a major topic in Stage 5 Mathematics. Students often need factorisation to solve quadratic equations, simplify expressions, and understand the connection between algebra and graphs.
Quadratics are also closely connected to parabolas. When students study parabolas, they need to understand how different forms of a quadratic expression show different information about the graph.
For example:
y = x² + 5x + 6
can be factorised as:
y = (x + 2)(x + 3)
This helps students find the x-intercepts of the parabola. That is why expanding and factorising are not just isolated algebra skills. They are foundation skills for Year 10 algebra, quadratic equations and parabola graphing.
How can a private maths tutor help my child with NSW Year 9 and Year 10 algebra, expansion and factorisation?
A private maths tutor can help by first checking whether the student understands the basics of algebra, brackets, multiplication and common factors.
A good tutor will not just give random questions. They will explain the full structure of the topic first, so the student understands what expansion and factorisation actually mean. Then they will teach each method step by step, including expanding single brackets, collecting like terms, factorising common terms and moving towards quadratic expressions when the student is ready.
After that, the student needs regular practice. Expansion and factorisation become easier when students see many examples and learn how to recognise the method needed for each question.
At Aussie Math Tutor NSW, we focus on clear explanations, step-by-step working and repeated practice so students can become more confident with algebra.
Do your practice tests cover the exact NSW syllabus?
Our practice questions are created to support the NSW Mathematics syllabus and are based on common school-style questions, past exam patterns and the skills students are expected to develop in Years 9 and 10.
The practice tests are designed to help students revise important skills such as expanding brackets, factorising common terms, expanding double brackets and factorising quadratic expressions.
However, schools may teach topics in a slightly different order, and the syllabus or assessment style may change over time. That is why students should also follow their school teacher’s guidance and use our practice tests as extra support for revision and confidence-building.
What is the difference between expanding and factorising?
Expanding means removing brackets by multiplying the term outside the bracket with each term inside the bracket.
Example:
4(x + 3) = 4x + 12
Factorising is the reverse process. It means taking out the common factor and putting the expression back into brackets.
Example:
4x + 12 = 4(x + 3)
A simple way to remember it is:
Expanding opens brackets. Factorising creates brackets.
Why do students find expanding and factorising difficult?
Many students struggle with expanding and factorising because they try to memorise steps without understanding what is happening.
Common mistakes include:
- multiplying only the first term inside the bracket
- forgetting negative signs
- not collecting like terms properly
- not recognising the highest common factor
- confusing expansion with factorisation
Once students understand that expanding and factorising are opposite processes, the topic becomes much easier. Regular practice also helps students recognise patterns faster.
Is factorising needed for quadratic equations and parabolas?
Yes. Factorising is very important for both quadratic equations and parabolas.
In quadratic equations, factorising helps students solve equations such as:
x² + 5x + 6 = 0
which becomes:
(x + 2)(x + 3) = 0
In parabolas, factorised form helps students find where the graph crosses the x-axis. This makes factorising a key skill for algebra and graphing in Stage 5 Mathematics.
