Expanding Brackets | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Expanding One Bracket

How do I expand a bracket?

• The expression 3x(x + 2) means 3x multiplied by the bracket (x + 2). 
  • Here, 3x is the term outside the bracket (referred to as a "factor"), and x + 2 are the terms inside the bracket.
• Expanding the brackets involves multiplying the term on the outside by each term on the inside, which eliminates the brackets.
  • 3x(x + 2) expands to 3x x x + 3x x 2, which simplifies to 3x² + 6.

Beware of minus signs

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

Expand & Simplify

How to simplify an expression with multiple terms in brackets?

• Identify two or more terms enclosed in brackets in the expression that are being added or subtracted. For instance, if you see two sets of brackets connected by a plus or minus sign, you are not multiplying the brackets together.
  • For example, consider the expression:
• Step 1: Expand each set of brackets individually by multiplying the term outside the brackets with each term inside. Pay attention to handling negative terms correctly.
  • E.g. the first set of brackets expands to 4 x x + 4 x 7, and simplifies to 4x + 28, the second set of brackets expands to  and simplifies to
  • 
• Step 2: Combine like terms after expanding the brackets.
  • 

Expanding Double Brackets

• Every term in the first bracket must be multiplied by every term in the second bracket. To expand (x + 1)(x + 3), you will need 4 multiplications in total.
• A good way to remember all the multiplications is using the acronym 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 is beneficial to enclose negative terms in brackets when multiplying.
• Simplify the final answer by combining like terms if present.

How do I expand two brackets using a grid?

• To expand (x + 1)(x + 3), represent the brackets as headings in a grid.
• 
• For each cell in the grid, multiply the term in the row heading by the term in the column heading.
• 
• Add all the terms inside the grid to calculate the answer.
• x² + x + 3x + 3
• Collect like terms to simplify the final expression.
• 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

Expanding Triple Brackets

How to Expand Three Brackets?

• Multiply out any two brackets using a standard method and simplify the answer by collecting like terms.
• Combine the two brackets into a single long bracket containing the expanded result.
• Proceed to expand this long bracket with the third (unused) bracket. 
  • The process typically resembles multiplying expressions like (x + a)(x² + bx + c).
  • Each term in the first bracket should be multiplied with every term in the second bracket, resulting in a total of six terms.
  • Using a grid can aid in keeping track of all six terms. 
    • 
    • For instance, for expressions like (x + 2)(x² + 3x + 1), adding terms in the grid (diagonals represent like terms) gives x³ + 2x² + 3x² + 6x + x + 2. 
    • Combine like terms to obtain the final answer of x³ + 5x² + 7x + 2.
• Ensure to simplify the final answer further by collecting any remaining like terms, putting negative terms in brackets when multiplying.