Expanding & Factorising Linear Expressions
Match Expression to Expanded Form
Draw a line to match each expression to its expanded form.
Expand the Brackets
Circle the correct expanded form.
3(x + 4)
5(2y − 3)
−2(a − 6)
4(3n + 1)
Match to Factorised Form
Draw a line to match each expression to its factorised form.
Factorise the Expression
Circle the correct factorised form.
8x + 12
15y − 10
6a + 9b
14m − 21
Expand and Simplify
Expand then collect like terms.
2(x + 3) + 3(x + 1)
4(y − 2) − 2(y + 1)
3(2a + 1) + 2(a − 3)
Fully Simplified or Not?
Sort: which expressions are already fully simplified?
Algebra in Context
Write and simplify expressions.
A rectangle has length (3x + 2) and width 4. Write and expand an expression for its perimeter.
Factorise fully: 12x² + 18x. Show the HCF and your working.
Expanding with Negative Coefficients
Expand each expression carefully — watch the sign of every term.
−3(x + 5)
−2(4a − 3)
−(6n − 1)
5(−2y + 3)
Factorising with Negative HCF
Factorise, taking out a negative HCF where shown.
−4x − 12
−6a + 9
−10m − 15
Substituting into Expanded Expressions
Substitute the given value and evaluate.
3(x + 4) when x = 2
−2(a − 5) when a = 1
4(2n + 1) when n = −1
5(y − 3) when y = 3
Perimeter and Area with Brackets
Write an expression with brackets, then expand it. Show all working.
A room has length (2x + 5) m and width 3 m. Write an expression for the perimeter using brackets, then expand it.
A border of tiles is placed around a rectangular garden that is x m by 4 m. The border adds 1 m on each side. Write an expression for the border area using brackets and expand it.
Expanding or Factorising?
Sort each task: are you expanding (removing brackets) or factorising (adding brackets)?
Algebra in Architecture
Explore how algebra describes real measurements.
- 1Measure the length and width of a room in your home. If one dimension is expressed as (2x + 1) metres, find the value of x. Calculate the perimeter using the expression and check against direct measurement.
- 2Design a rectangular garden where one side is 3 m longer than twice the other side. Write an expression for the perimeter. If the shorter side is 4 m, what is the perimeter?
- 3Look at a product in a supermarket with a price of $(3n + 50) cents. What value of n makes this equal to $2.00? Factorise the expression.
Three-Term Expansion
Expand the bracket — it has three terms inside.
2(x + y + 3)
3(a − 2b + 4)
−4(2m + n − 1)
5(x + 2y − z)
Find the HCF — Sort by Value
Identify the correct HCF for each expression and sort from smallest to largest HCF.
Expand and Simplify — Two Brackets
Expand both brackets, then collect like terms.
3(x + 2) + 2(x + 5)
4(y − 1) − 2(y + 3)
5(2a + 1) − 3(a − 2)
2(3m − 4) + 3(m + 2)
Collect Like Terms After Expanding
Expand and fully simplify each expression. Show all working.
Simplify: 4(2x + 3) − 2(x − 1)
Simplify: 3(a + b) + 2(2a − 3b)
Simplify: 5(y + 4) − 3(y + 6) + 2(y − 1)
Algebraic Word Problems
Write an algebraic expression with brackets, then expand. Show all working.
A rectangle has length (3x + 4) cm and width 5 cm. Write and expand an expression for its perimeter.
A square has side length (2m − 1) cm. Write an expression for its area and expand it.
Three friends each have (n + 7) dollars. Write and simplify an expression for their total money.
Which Factorisation is Complete?
Circle the fully factorised form.
12x + 18:
8y − 12:
15a + 10b:
6x² + 9x:
Factorising — Challenging Problems
Factorise each expression completely. Show the HCF at each step.
Factorise: 24a²b − 16ab²
Factorise: 3x(y + 2) + 6(y + 2). (Hint: (y + 2) is a common factor.)
Factorise: x(x + 3) − 2(x + 3). Simplify fully.
Sort by Factorisation Method
Sort each expression by what type of factorisation it requires.
Expanding Brackets — Using Algebra to Solve
Expand and use the resulting expression to solve.
A rectangle has width w and length (w + 5). Its perimeter is 34 cm. Write and expand the perimeter expression, then solve for w.
Two rectangles: Rectangle A is 3(x + 2) cm² and Rectangle B is 2(x + 4) cm². They have equal areas. Solve for x.
Introduction to Expanding Two Brackets (FOIL)
Use FOIL (First, Outside, Inside, Last) to expand each product of two brackets.
Expand (x + 2)(x + 5). F = x × x = ___, O = x × 5 = ___, I = 2 × x = ___, L = 2 × 5 = ___. Collect like terms.
Expand (x + 3)(x − 4). Show full FOIL working and simplify.
Expand (2x + 1)(x + 3). Show full working.
Expanding with Negatives — Set A
Expand. Be careful with signs.
−3(x − 4)
−(2x + 5y − 1)
2(3x + 1) − (x − 5)
3(2x − y) + 2(x + 4y)
Factorising — Set A
Factorise fully by taking out the highest common factor (HCF).
6x + 9
12y² − 8y
5x² + 15x
−4a + 10b − 6
Verify by Expanding
Expand the factorised form to check if it equals the original.
Is 3(2x + 5) = 6x + 15?
Is 4(x − 2) + 3 = 4x − 5?
Is 2(3x + 4) − (x + 2) = 5x + 6?
Expanding — Set B
Expand and collect like terms.
3(2x + 5) + 4(x − 1)
2(3y − 4) − 3(y + 2)
x(x + 3) + 2(x² − 1)
5(2a − b) − (a + 3b)
Factorising — Set B
Factorise fully.
14x² − 21xy
8a²b + 12ab²
15p³ − 25p²q + 10pq
−6x² + 9x
Expand or Factorise?
Sort each task: does it require expanding or factorising?
Algebra in Geometry
Use expanding or factorising to solve each geometry problem.
A rectangle has length (2x + 5) cm and width 3 cm. Write an expression for the area. Expand and simplify.
The perimeter of a square is (8x + 12) cm. Write an expression for the side length by factorising.
Two rectangles are placed side by side. One has dimensions 4 × x and the other has dimensions 4 × 3. Write the total area by factorising.
Solving Equations by Expanding First
Expand brackets, then solve the equation.
3(x + 4) = 21. Expand, then solve.
2(3x − 1) = 4(x + 3). Expand both sides, then solve.
5(2x + 1) − 3(x − 2) = 28. Expand, simplify, then solve.
Algebraic Expressions — Word Problems
Write and simplify an algebraic expression for each problem.
A box contains n chocolates. Alice takes 3 handfuls of n chocolates and Bob takes 2 handfuls of (n − 4). Write and simplify an expression for the total taken.
The cost of a party venue is $50 per person for adults and $30 per person for children. There are a adults and c children. Write an expression for the total cost. If 8 adults and 12 children attend, find the total.
Factorising to Simplify
Factorise the numerator and cancel common factors to simplify each fraction.
Simplify (6x² + 9x) ÷ 3x
Simplify (12y − 8) ÷ 4
Simplify (10a²b + 15ab²) ÷ 5ab
Error Analysis
Find and fix the errors.
Student: '−2(x − 3) = −2x − 6.' What is wrong? Write the correct expansion.
Student: 'The HCF of 8x² and 12x is 4, so 8x² + 12x = 4(2x² + 3x).' Is this fully factorised? How could it be improved?
Creating Equivalent Expressions
For each expression, write two equivalent forms: one expanded and one factorised.
Start with 3x(4 − 2x). Expanded form? Factorised form?
Start with 5y + 15. Factorised form? Check by expanding.
Reflection: Expanding and Factorising
Summarise what you have learned.
Explain expanding in your own words with one example.
Explain factorising in your own words with one example.
When would you expand an expression? When would you factorise? Give a real example of each.