National 5MathsSQARevision NotesAlgebraic SkillsExpansion of BracketsSingle Brackets

Single Brackets (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Expanding single brackets

How do I expand single brackets?

  • The expression 3 x left parenthesis x plus 2 right parenthesis means 3 x  multiplied by the bracket left parenthesis x plus 2 right parenthesis

    • 3 x is the term outside the bracket

      • this is sometimes called a factor

    • and x plus 2 are the terms inside the bracket

  • Expanding the brackets means multiplying the outside term by each term on the inside

    • This will remove (get rid of) the brackets

    • 3 x left parenthesis x plus 2 right parenthesis expands to 3 x cross times x plus 3 x cross times 2 which simplifies to 3 x squared plus 6 x

  • Beware of minus signs

    • Remember the rules
      −  ×  −  =  +
      −  ×  +  =  − 

    • It helps to put brackets around negative terms

Worked Example

(a) Expand  4 x open parentheses 2 x minus 3 close parentheses.

(b) Expand  negative 7 x open parentheses 4 minus 5 y close parentheses.

Answer:

Part (a)

 Multiply the 4 x term outside the brackets by both terms inside the brackets

4 x cross times 2 x plus 4 x cross times open parentheses negative 3 close parentheses

Carry out the multiplications

8 x squared minus 12 x

Part (b)

Multiply the negative 7 x outside the brackets by both terms inside the brackets

open parentheses negative 7 x close parentheses cross times 4 plus open parentheses negative 7 x close parentheses cross times open parentheses negative 5 y close parentheses

Carry out the multiplications

  • Remember that multiplying two negatives gives a positive

negative 28 x plus 35 x y

Expand & simplify

How do I simplify brackets that are added together?

  • First expand both brackets separately

    • 4 open parentheses x plus 7 close parentheses plus 5 x open parentheses 3 minus x close parentheses 

      • The first set of brackets expands to 4 cross times x plus 4 cross times 7 which simplifies to 4 x plus 28

      • The second set of brackets expands to 5 x cross times 3 plus 5 x cross times open parentheses negative x close parentheses which simplifies to 15 x minus 5 x squared

      • So 4 open parentheses x plus 7 close parentheses plus 5 x open parentheses 3 minus x close parentheses equals 4 x plus 28 plus 15 x minus 5 x squared

  • Then collect like terms

    • 4 x plus 15 x equals 19 x

      • The other two terms are not like terms

    • So 4 open parentheses x plus 7 close parentheses plus 5 x open parentheses 3 minus x close parentheses equals 19 x plus 28 minus 5 x squared 

Worked Example

(a) Expand and simplify  2 open parentheses x plus 5 close parentheses plus 3 x open parentheses x minus 8 close parentheses.

(b) Expand and simplify  3 x open parentheses x plus 2 close parentheses minus 7 open parentheses x minus 6 close parentheses.

Answer:

Part (a)

Expand each set of brackets separately

You can keep negative terms inside brackets

2 cross times x plus 2 cross times 5 plus 3 x cross times x plus 3 x cross times open parentheses negative 8 close parentheses

Carry out the multiplications

2 x plus 10 plus 3 x squared minus 24 x

Collect like terms (the 2x and the -24x)

negative 22 x plus 10 plus 3 x squared

Part (b)

Expand each set of brackets separately

  • Be careful: the second set of brackets has a -7 in front, not +7

 3 x cross times x plus 3 x cross times 2 plus open parentheses negative 7 close parentheses cross times x plus open parentheses negative 7 close parentheses cross times open parentheses negative 6 close parentheses

Carry out the multiplications

  • Remember that multiplying two negatives gives a positive

3 x squared plus 6 x minus 7 x plus 42

Collect like terms

3 x squared minus x plus 42

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.