How to Expand and Factorise Brackets in Algebra

Expanding and factorising are two of the most important skills in algebra. They are opposite operations: expanding removes brackets, and factorising puts them back in. You will use both constantly from Year 8 through to Year 12 and beyond.

This guide covers expanding single brackets, expanding double brackets, and factorising common factors and simple quadratics. Each section builds on the one before it.

Expanding single brackets

To expand a single bracket, multiply the term outside the bracket by every term inside the bracket.

Example 1

Expand 3(x + 4).

Multiply 3 by x: 3x

Multiply 3 by 4: 12

Result: 3x + 12

Example 2

Expand -2(5y - 3).

Multiply -2 by 5y: -10y

Multiply -2 by -3: +6 (negative times negative is positive)

Result: -10y + 6

The sign errors in that second example are exactly where most students trip up. When the term outside the bracket is negative, you must be careful with every multiplication. It helps to write out each step rather than trying to do it in your head.

Example 3

Expand x(x - 7).

Multiply x by x: x²

Multiply x by -7: -7x

Result: x² - 7x

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Expanding double brackets (FOIL)

When you have two brackets multiplied together, you need to multiply every term in the first bracket by every term in the second bracket. The FOIL method gives you a systematic way to do this:

• First: multiply the first terms of each bracket
• Outer: multiply the outer terms
• Inner: multiply the inner terms
• Last: multiply the last terms of each bracket

Then combine like terms.

Example 1

Expand (x + 3)(x + 5).

F: x times x = x²

O: x times 5 = 5x

I: 3 times x = 3x

L: 3 times 5 = 15

Combine: x² + 5x + 3x + 15 = x² + 8x + 15

Example 2

Expand (2x - 1)(x + 4).

F: 2x times x = 2x²

O: 2x times 4 = 8x

I: -1 times x = -x

L: -1 times 4 = -4

Combine: 2x² + 8x - x - 4 = 2x² + 7x - 4

Example 3

Expand (x - 6)(x - 2).

F: x times x = x²

O: x times -2 = -2x

I: -6 times x = -6x

L: -6 times -2 = 12

Combine: x² - 2x - 6x + 12 = x² - 8x + 12

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Factorising: taking out a common factor

Factorising is the reverse of expanding. The first type you will learn is taking out the highest common factor (HCF) of all terms.

The method

1. Find the HCF of all the terms (both numbers and variables).
2. Write the HCF outside a bracket.
3. Inside the bracket, write what is left when you divide each term by the HCF.

Example 1

Factorise 6x + 18.

HCF of 6x and 18 is 6.

6x / 6 = x

18 / 6 = 3

Result: 6(x + 3)

Check by expanding: 6(x + 3) = 6x + 18. Correct.

Example 2

Factorise 4x² - 10x.

HCF of 4x² and 10x is 2x.

4x² / 2x = 2x

10x / 2x = 5

Result: 2x(2x - 5)

Example 3

Factorise 3a²b + 9ab².

HCF of 3a²b and 9ab² is 3ab.

3a²b / 3ab = a

9ab² / 3ab = 3b

Result: 3ab(a + 3b)

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Factorising quadratics

A quadratic expression like x² + 8x + 15 can be factorised into two brackets. You need to find two numbers that multiply to give the constant term (15) and add to give the coefficient of x (8).

Example

Factorise x² + 8x + 15.

Find two numbers that multiply to 15 and add to 8.

Try: 3 and 5. Check: 3 × 5 = 15 and 3 + 5 = 8. Yes.

Result: (x + 3)(x + 5)

You can verify this is correct by expanding (x + 3)(x + 5) using FOIL. You should get back to x² + 8x + 15.

When the constant is negative

Factorise x² - 8x + 12.

Find two numbers that multiply to 12 and add to -8.

Since the product is positive and the sum is negative, both numbers must be negative.

Try: -6 and -2. Check: (-6) × (-2) = 12 and (-6) + (-2) = -8. Yes.

Result: (x - 6)(x - 2)

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Common mistakes to avoid

• Sign errors when expanding: Especially with negative terms outside brackets. Write every step out.
• Forgetting to multiply by every term: In 3(x + 4), some students write 3x + 4 instead of 3x + 12. The 3 must multiply both terms.
• Not taking out the full HCF: In 4x² - 10x, taking out just 2 gives 2(2x² - 5x), which is not fully factorised. You should take out 2x.
• Not checking by expanding: After factorising, expand your answer to make sure you get back to the original expression. This takes 30 seconds and catches most errors.

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Building from here

Expanding and factorising are not just standalone skills. They are tools you will use to solve equations, simplify algebraic fractions, complete the square, and work with the quadratic formula. The more fluent you are with these operations, the easier all of those later topics become.

If you want structured practice with expanding and factorising, imSteyn covers algebra across Years 8 to 10 in the Australian Curriculum. It guides you through each type of problem step by step, asking questions along the way so you build understanding rather than just repeating procedures.