Expanding Brackets (Cambridge O Level Maths)

How do I expand a bracket?

• The expression 3x(x + 2) means 3x  multiplied by the bracket (x + 2)
  • 3x is the term outside the bracket (sometimes called a "factor") and x + 2 are the terms inside the bracket
• Expanding the brackets means multiplying the term on the outside by each term on the inside
  • This will remove / "get rid of" the brackets
  • 3x(x + 2) expands to which simplifies to 

Beware of minus signs

• Remember the basic rules of multiplication with signs
  •  −  ×  −  =  +
  • −  ×  +  =  − 
• It helps to put brackets around negative terms

How do I simplify an expression when there is more than one term in brackets?

• Look out for two or more terms that contain brackets in an expression that are being added/subtracted
  • E.g.
  • Notice that the two sets of brackets are connected by a + sign, so you are not multiplying the brackets together
• STEP 1: Expand each set of brackets separately by multiplying the term on the outside of the brackets by each of the terms on the inside, be careful with negative terms
  • E.g. the first set of brackets expands to , and simplifies to , the second set of brackets expands to  and simplifies to 
  • So,
• STEP 2: Collect together like terms
  • E.g.  

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

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