Expanding Multiple Brackets (OCR GCSE Maths)

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Mark

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Mark

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Expanding Double Brackets

How do I expand two brackets using FOIL?

  • Every term in the first bracket must be multiplied by every term in the second bracket
    • To expand (x + 1)(x + 3) will need 4 multiplications in total
  • A good way to remember all the multiplications is FOIL
    • F = First: multiply together the first terms in each bracket
    • O = Outside: multiply the first term in the first bracket by the second term in the second bracket (visually, these are the "outer" terms)
    • I = Inside: multiply the second term in the first bracket by the first term in the second bracket (visually, these are the "inner" terms)
    • L = Last: multiply together the last terms in each bracket

  • It helps to put negative terms in brackets when multiplying
  • Simplify the final answer by collecting like terms (if there are any)

 

How do I expand two brackets using a grid?

  • To expand (x + 1)(x + 3), write out the brackets as headings in a grid (in either direction)
  •   x +1
    x    
    +3    
  • For each cell in the middle, multiply the term in the row heading by the term in the column heading
  •   x +1
    x x2 x
    +3 3x 3
  • Add together all the terms inside the grid to get the answer
    • x2 + x + 3x + 3
    • collect like terms
    • x2 + 4x + 3

 

How do I expand a bracket squared?

  • Write (x + 3)2 as (x + 3)(x + 3) and use one of the methods above
    • for example, with FOIL: (x + 3)(x + 3) = x2 + 3x + 3x + 9
    • collect like terms to get the final answer
    • x2 + 6x + 9
  • Do not make the common mistake of saying (x + 3)2 is x2 + 32
    • This cannot be true, as substituting x = 1, for example, gives (1 + 3)2 = 42 = 16 on the left, but 12 + 32 = 1 + 9 = 10 on the right

Worked example

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

 

Using FOIL, multiply together the first, outer, inner and last terms

space space space space space space space straight F space space space space space space space space space space space space space space space space space space straight O space space space space space space space space space space space space space space space space space space space space space straight I space space space space space space space space space space space space space space space space space space space space space space space straight L
circle enclose 2 x cross times x end enclose plus circle enclose 2 x cross times 4 end enclose plus circle enclose open parentheses negative 3 close parentheses cross times x end enclose plus circle enclose open parentheses negative 3 close parentheses cross times 4 end enclose

Simplify each term

2 x squared plus 8 x minus 3 x minus 12

Collect like terms (the 8x and -3x)

bold 2 bold italic x to the power of bold 2 bold plus bold 5 bold italic x bold minus bold 12

 

(b)
Expand  open parentheses x minus 3 close parentheses open parentheses 3 x minus 5 close parentheses.

 

Using FOIL, multiply together the first, outer, inner and last terms

space space space space space space space straight F space space space space space space space space space space space space space space space space space space space space straight O space space space space space space space space space space space space space space space space space space space space space space space space space straight I space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight L
circle enclose x cross times 3 x end enclose plus circle enclose x cross times open parentheses negative 5 close parentheses end enclose plus circle enclose open parentheses negative 3 close parentheses cross times 3 x end enclose plus circle enclose open parentheses negative 3 close parentheses cross times open parentheses negative 5 close parentheses end enclose

Simplify each term

3 x squared minus 5 x minus 9 x plus 15

Collect like terms (the -5x and -9x)

bold 3 bold italic x to the power of bold 2 bold minus bold 14 bold italic x bold plus bold 15

Worked example

Expand  open parentheses 2 x plus 3 close parentheses squared.


Rewrite the expression as two separate brackets multiplied together.

open parentheses 2 x plus 3 close parentheses open parentheses 2 x plus 3 close parentheses

Using FOIL, multiply together: the first terms, the outside terms, the inside terms and the last terms.

space space space space space space space space space straight F space space space space space space space space space space space space space space space space space space space space straight O space space space space space space space space space space space space space space space space straight I space space space space space space space space space space space space space space space space space straight L
circle enclose 2 x cross times 2 x end enclose plus circle enclose 2 x cross times 3 end enclose plus circle enclose 3 cross times 2 x end enclose plus circle enclose 3 cross times 3 end enclose

Simplify.

4 x squared plus 6 x plus 6 x plus 9

Collect 'like' terms.

bold 4 bold italic x to the power of bold 2 bold plus bold 12 bold italic x bold plus bold 9

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Expanding Triple Brackets

How do I expand three brackets?

  • Multiply out any two brackets using a standard method and simplify this answer (collect any like terms)
  • Replace the two brackets above with one long bracket containing the expanded result
  • Expand this long bracket with the third (unused) bracket
    • This step often looks like (x + a)(x2 + bx + c)
    • Every term in the first bracket must be multiplied with every term in the second bracket
      • This leads to six terms 
    • A grid can often help to keep track of all six terms, for example (x + 2)(x2 + 3x + 1)
      •   x2 +3x +1
        x x3 3x2

        x

        +2 2x2 6x 2
      • add all the terms inside the grid (diagonals show like terms) to get x3 + 2x2 + 3x2 + 6x + x + 2
      • collect like terms to get the final answer of x3 + 5x2 + 7x + 2
  • Simplify the final answer by collecting like terms (if there are any)
  • It helps to put negative terms in brackets when multiplying

Worked example

(a)
Expand  open parentheses 2 x minus 3 close parentheses open parentheses x plus 4 close parentheses open parentheses 3 x minus 1 close parentheses.


Start by expanding the first two sets of brackets using the FOIL method and simplify by collecting 'like' terms.

table row blank blank cell open parentheses 2 x minus 3 close parentheses open parentheses x plus 4 close parentheses end cell row blank equals cell 2 x cross times x plus 2 x cross times 4 plus open parentheses negative 3 close parentheses cross times x plus open parentheses negative 3 close parentheses cross times 4 end cell row blank equals cell 2 x squared plus 8 x minus 3 x minus 12 end cell row blank equals cell 2 x squared plus 5 x minus 12 end cell end table

Rewrite the original expression with the first two brackets expanded.

open parentheses 2 x squared plus 5 x minus 12 close parentheses open parentheses 3 x minus 1 close parentheses

Multiply all of the terms in the first set of brackets by all of the terms in the second set of brackets.

2 x squared cross times 3 x plus 5 x cross times 3 x plus open parentheses negative 12 close parentheses cross times 3 x plus 2 x squared cross times open parentheses negative 1 close parentheses plus 5 x cross times open parentheses negative 1 close parentheses plus open parentheses negative 12 close parentheses cross times open parentheses negative 1 close parentheses

Simplify.

6 x cubed plus 15 x squared minus 36 x minus 2 x squared minus 5 x plus 12

Collect 'like' terms.

bold 6 bold italic x to the power of bold 3 bold plus bold 13 bold italic x to the power of bold 2 bold minus bold 41 bold italic x bold plus bold 12

 

(b)
Expand  open parentheses x minus 3 close parentheses open parentheses x plus 2 close parentheses open parentheses 2 x minus 1 close parentheses.

Start by expanding the first two sets of brackets using the FOIL method and simplify by collecting 'like' terms.

table row blank blank cell open parentheses x minus 3 close parentheses open parentheses x plus 2 close parentheses end cell row blank equals cell x cross times x plus x cross times 2 plus open parentheses negative 3 close parentheses cross times x plus open parentheses negative 3 close parentheses cross times 2 end cell row blank equals cell x squared plus 2 x minus 3 x minus 6 end cell row blank equals cell x squared minus x minus 6 end cell end table

Rewrite the original expression with the first two brackets expanded.

open parentheses x squared minus x minus 6 close parentheses open parentheses 2 x minus 1 close parentheses

Multiply all of the terms in the first set of brackets by all of the terms in the second set of brackets.

x squared cross times 2 x plus open parentheses negative x close parentheses cross times 2 x plus open parentheses negative 6 close parentheses cross times 2 x plus x squared cross times open parentheses negative 1 close parentheses plus open parentheses negative x close parentheses cross times open parentheses negative 1 close parentheses plus open parentheses negative 6 close parentheses cross times open parentheses negative 1 close parentheses

Simplify.

2 x cubed minus 2 x squared minus 12 x minus x squared plus x plus 6

Collect 'like' terms.

bold 2 bold italic x to the power of bold 3 bold minus bold 3 bold italic x to the power of bold 2 bold minus bold 11 bold italic x bold plus bold 6

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.