Practice Maths

Factorising Algebraic Expressions

Key Ideas

factor
A number or expression that divides evenly into another — 3 and (2x + 1) are both factors of 3(2x + 1).
highest common factor (HCF)
The largest factor that divides every term in an expression — the HCF of 12x2 and 8x is 4x.
factorise
Rewrite an expression as a product of factors — the reverse of expanding; e.g. 12x2 + 8x = 4x(3x + 2).
coefficient
The numerical part of a term — in −6x2, the coefficient is −6.

Finding the HCF

For numbers: list factors and find the largest shared one.   For variables: use the lowest power that appears in every term.
e.g. HCF of 12x² and 8x — number HCF of 12 and 8 is 4; variable HCF is x (lowest power) → HCF = 4x.

Hot Tip After factorising, always expand your answer to check. If you get the original expression back, your factorisation is correct!

Worked Example

Question: Factorise 12x² + 8x.

Step 1 — Find the HCF of all terms.
HCF of 12 and 8 is 4. HCF of x² and x is x. So HCF = 4x.

Step 2 — Write the HCF outside the brackets.
12x² + 8x = 4x(     )

Step 3 — Divide each term by the HCF to fill the brackets.
12x² ÷ 4x = 3x    and    8x ÷ 4x = 2
So: 12x² + 8x = 4x(3x + 2)

Step 4 — Check by expanding.
4x(3x + 2) = 12x² + 8x ✓

Factorising: The Reverse of Expanding

Expanding takes a product and turns it into a sum: 3(x + 4) = 3x + 12. Factorising does the reverse: it takes a sum and turns it into a product: 3x + 12 = 3(x + 4). This is like "packaging" the expression back into a compact bracket form. Why do we do this? Factorised forms are often easier to work with when solving equations, simplifying fractions, or finding when an expression equals zero.

To factorise by taking out the highest common factor (HCF): first find the largest number and the highest power of any variable that divides every term. For 12x2 + 8x: HCF of 12 and 8 is 4 (not 2, not 1 — we want the highest). HCF of x2 and x is x (the lowest power present). So HCF = 4x. Write 4x( ), then fill the bracket by dividing each original term by 4x: 12x2 ÷ 4x = 3x and 8x ÷ 4x = 2. Result: 4x(3x + 2).

Finding the HCF Systematically

For numbers: list all factors of each coefficient and pick the largest shared one. For 18 and 24: factors of 18 are 1, 2, 3, 6, 9, 18; factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Largest shared factor is 6. For variables: take the lowest power that appears in every term. For x3, x2, x: lowest power is x1 = x. For x4 and x2: lowest is x2.

Example: factorise 18x3 − 24x2. HCF of 18 and 24 = 6. HCF of x3 and x2 = x2. So HCF = 6x2. Write 6x2(___). Fill in: 18x3 ÷ 6x2 = 3x and 24x2 ÷ 6x2 = 4. Result: 6x2(3x − 4).

Key Tip: Always use the HIGHEST common factor, not just any common factor. If you factor out 2 from 12x2 + 8x, you get 2(6x2 + 4x) — which is partially factorised but not fully. The expression inside still has a common factor of 2x. Fully factorised means nothing more can be taken out.

Always Check by Expanding

After factorising, expand your answer to check it equals the original expression. If 6x2(3x − 4): expand to get 18x3 − 24x2 — matches the original. This check takes 10 seconds and guarantees your factorisation is correct. Make it a habit.

With a negative leading term like −6x + 15: factor out −3 (it's conventional to make the bracket have a positive leading term): −3(2x − 5). Check: −3 × 2x = −6x and −3 × (−5) = +15. Matches original!

Key Tip: Factorising always comes in two parts: (1) finding the HCF, and (2) dividing every term by it. Don't guess the bracket contents — calculate them by division. And always verify by expanding back.

Why Factorising Matters

Factorising is one of the most important algebra skills you'll use throughout high school and beyond. In Year 9 you'll factorise quadratics (trinomials). In Year 10 and 11 you'll use it to solve equations and simplify rational expressions. Factorising also appears in physics (resolving forces), economics (break-even analysis), and computer science. Mastering HCF factorisation now builds the foundation for all of these skills.

Mastery Practice

  1. Factorise each expression by taking out the numeric HCF. Fluency

    1. 6x + 9
    2. 10y − 15
    3. 4a + 12
    4. 8m − 20
    5. 14p + 21
    6. 6t − 18
    7. 15n + 25
    8. 12k − 30

  2. Factorise each expression by taking out the variable HCF. Fluency

    1. x² + 5x
    2. 3y² − 7y
    3. 4a² + 6a
    4. 10m² − 15m
    5. 8p² + 12p
    6. 6t² − 9t
    7. 5n² + 20n
    8. 12k² − 8k

  3. Factorise each expression fully (numeric and variable HCF). Understanding

    1. 6x² + 9x
    2. 10y² − 15y
    3. 4a² + 8a
    4. 12m² − 18m
    5. 15p² + 25p
    6. 9b² − 6b

  4. Factorise each three-term expression, then verify by expanding. Understanding

    1. 6x² + 9x + 3
    2. 4y² − 8y + 12
    3. 10a + 15a² − 5
    4. 12m² + 6m − 18
    5. 8t² + 4t + 16
    6. Expand 5(2k − 3), then re-factorise your expansion to check you get back 5(2k − 3).

  5. Factorise each expression by taking out a negative factor. Understanding

    1. −4x − 8 (factorise as −4(...))
    2. −6y + 9
    3. −3a² − 12a
    4. −10m² + 15m

  6. Apply factorising to solve problems. Problem Solving

    1. A rectangle has area = 8x + 12 cm². Write two possible sets of dimensions (length × width) using factorising. Which factorisation uses the HCF?
    2. A gardener designs a garden bed with area 6x² + 10x m². Factorise this expression and explain what each factor could represent about the shape of the garden.
    3. A student writes: “The HCF of 6x² + 4x is 2.” Is this correct? Explain why or why not, and give the fully factorised form.
    4. The perimeter of an equilateral triangle is (15x + 9) cm. Write an expression for one side length, fully simplified.

  7. Factorise each expression fully. The HCF is larger than 2 in each case. Fluency

    1. 18x + 27
    2. 24y − 36
    3. 15a + 35
    4. 21m − 28
    5. 16p + 40
    6. 45n − 30
    7. 12k + 48
    8. 32b − 24

  8. Factorise each three-term expression by finding the HCF of all three terms. Fluency

    1. 3x + 6y + 9
    2. 4a − 8b + 12
    3. 5m + 10n − 15
    4. 6p² + 9p + 3
    5. 10x² − 5x + 15
    6. 8a² + 4a − 12

  9. Each expression below is already factorised. Expand it to verify, then re-factorise using the HCF. Understanding

    1. 3(4x + 5)
    2. 2y(3y − 7)
    3. 5(a2 + 2a − 3)
    4. 4m(2m + 5)
    5. 6p(p − 3)
    6. −2(3n + 8)

  10. Use factorising to solve real-world and algebraic problems. Problem Solving

    1. A farmer divides a field of area (12x + 20) m² into equal plots. What is the maximum number of equal plots possible? Write an expression for the area of each plot.
    2. A catering company prepares two types of sandwiches: 14p cheese sandwiches and 21p ham sandwiches. Factorise the total expression and explain what the factorised form tells you about how sandwiches are grouped.
    3. The area of a rectangle is (20x² − 12x) cm². Find two possible pairs of dimensions. Which pair uses the HCF?
    4. Two expressions are given: Expression A = 3(2x + 4) and Expression B = 6x + 12. Are they equivalent? Fully factorise both and explain your answer.
See Answers ➔