National 5 Maths
Expanding Brackets

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Course content

  • Expanding algebraic expressions of these four types:
    1.    \(a(bx+c)+d(ex+f)\)
    2.    \(ax(bx+c)\)
    3.    \((ax+b)(cx+d)\)
    4.    \((ax+b)(cx^2+dx+e)\)
    Note: \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) will always be integers
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Integer

The set of integers is the set of whole numbers {0,1,2,3,...} together with their negative counterparts {-1,-2,-3,...}.

Key ideas

  • For types 3 and 4 above, each term in the first bracket is multiplied by each term in the second bracket.
  • Take care with \(+\) and \(-\) signs:
    • positive \(\times\) negative \(=\) negative
    • negative \(\times\) negative \(=\) positive
  • We should always collect like terms to simplify our answers fully.

Textbook page references

Self-study course

  • National5.com   Highly recommended. Save £10 with discount code 'Maths.scot'.

Example 1 (Type 1)

Multiply out the brackets and collect like terms: \(5(x+4)+3(2x+1)\)

Example 2 (Type 1)

Multiply out the brackets and collect like terms: \(-2(3x+4)+5(2x-5)\)

Example 3 (Type 2)

Expand \(x(3x + 4)\)

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Example 4 (Type 2)

Expand \(6x(-2x + 3)\)

Example 5 (Type 3)

Expand the brackets and collect like terms: \((x+2)(x+5)\)

Example 6 (Type 3)

Expand the brackets and collect like terms: \((-x+6)(3x-5)\)

Recommended student books

Zeta Maths: National 5+ practice book 
Leckie: National 5 Maths textbook 

Example 7 (Type 4)

Expand and simplify fully: \((x+2)(x^2+x+3)\)

Example 8 (Type 4)

Expand and simplify fully: \((2x-1)(7x^2+x-5)\)

Example 9 (non-calculator)

SQA National 5 Maths 2014 P1 Q2

Expand the brackets and collect like terms: \((2x-5)(3x+1)\)

Recommended revision guides

How to Pass National 5 Maths 
BrightRED N5 Maths Study Guide 

Example 10 (non-calculator)

SQA National 5 Maths 2015 P1 Q4

Multiply out the brackets and collect like terms: \((x-4)(x^2+x-2)\)

Example 11 (non-calculator)

SQA National 5 Maths 2017 P1 Q4

Expand and simplify \((2x+3)(x^2-4x+1)\)

Example 12 (non-calculator)

SQA National 5 Maths 2018 P1 Q4

Expand and simplify \((3x+1)(x-1)+2(x^2-5)\)

N5 Maths practice papers

Non-calculator papers and solutions 
Calculator papers and solutions 

Example 13 (non-calculator)

SQA National 5 Maths 2019 P1 Q3

Expand and simplify \((x+5)(2x^2-7x-3)\)

Example 14 (non-calculator)

SQA National 5 Maths 2023 P1 Q2

Expand and simplify \((x+7)^2+6(x^2-10)\)

Example 15 (non-calculator)

SQA National 5 Maths 2024 P1 Q3

Expand and simplify \((x+1)(x^2-4x+5)\)

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Maths.scot worksheet

Expanding brackets worksheet
Answer sheet
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Past paper questions

All past paper questions by topic
Routine expansion:
2014 Paper 1 Q2
2015 Paper 1 Q4
2017 Paper 1 Q4
2018 Paper 1 Q2
2019 Paper 1 Q3
2021 Paper 1 Q3
2022 Paper 2 Q1
2023 Paper 1 Q2
2024 Paper 1 Q3
Within other topics:
2013 Spec. P1 Q7 (with indices)
2016 Paper 1 Q12 (with quadratics)
2017 Spec. P2 Q16 (with quadratics)
2024 Paper 1 Q13 (with indices)

Other great resources

Video - Larbert High School
Video - Tynecastle HS
Includes expanding three brackets
Video - LHS Maths
Videos - Clelland Maths
1. Single brackets
2. Double brackets
Video - DLB Maths
SQA 2014 Paper 1 Q2 solution
PowerPoint - MathsRevision.com
Lesson notes - Maths 777
1. Expanding and simplifying
2. Pair of brackets: both linear
3. Linear times quadratic
4. Expanding brackets: all types
Notes - BBC Bitesize
Test yourself - BBC Bitesize
Notes - National5.com
Notes - MathCentre.ac.uk
Essential Skills worksheets
Linear times linear (Answers)
Linear times quadratic (Answers)
Practice questions - Maths Hunter
Worksheet - St Andrew's Academy
Exercise on page 6
Worksheet - Airdrie Academy
Exercises - Larkhall Academy
Pages 21-25 Ex 2-6 (no answers)
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