Expanding Multiple Brackets (AQA GCSE Maths)

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Mark

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12 April 2024

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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

(2 x - 3) (x + 4)

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

$F O I L 2 x \times x + 2 x \times 4 + (- 3) \times x + (- 3) \times 4$

Simplify each term

$2 x^{2} + 8 x - 3 x - 12$

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

$2 x^{2} + 5 x - 12$

(b)

Expand

(x - 3) (3 x - 5)

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

$F O I L x \times 3 x + x \times (- 5) + (- 3) \times 3 x + (- 3) \times (- 5)$

Simplify each term

$3 x^{2} - 5 x - 9 x + 15$

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

$3 x^{2} - 14 x + 15$

Worked example

Expand ${(2 x + 3)}^{2}$ .

Rewrite the expression as two separate brackets multiplied together.

$(2 x + 3) (2 x + 3)$

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

$F O I L 2 x \times 2 x + 2 x \times 3 + 3 \times 2 x + 3 \times 3$

Simplify.

$4 x^{2} + 6 x + 6 x + 9$

Collect 'like' terms.

$4 x^{2} + 12 x + 9$

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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

(2 x - 3) (x + 4) (3 x - 1)

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

$\begin{array}{rcl} (2 x - 3) (x + 4) \\ = & 2 x \times x + 2 x \times 4 + (- 3) \times x + (- 3) \times 4 \\ = & 2 x^{2} + 8 x - 3 x - 12 \\ = & 2 x^{2} + 5 x - 12 \end{array}$

Rewrite the original expression with the first two brackets expanded.

$(2 x^{2} + 5 x - 12) (3 x - 1)$

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

$2 x^{2} \times 3 x + 5 x \times 3 x + (- 12) \times 3 x + 2 x^{2} \times (- 1) + 5 x \times (- 1) + (- 12) \times (- 1)$

Simplify.

$6 x^{3} + 15 x^{2} - 36 x - 2 x^{2} - 5 x + 12$

Collect 'like' terms.

$6 x^{3} + 13 x^{2} - 41 x + 12$

(b)

Expand

(x - 3) (x + 2) (2 x - 1)

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

$\begin{array}{rcl} (x - 3) (x + 2) \\ = & x \times x + x \times 2 + (- 3) \times x + (- 3) \times 2 \\ = & x^{2} + 2 x - 3 x - 6 \\ = & x^{2} - x - 6 \end{array}$

Rewrite the original expression with the first two brackets expanded.

$(x^{2} - x - 6) (2 x - 1)$

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

$x^{2} \times 2 x + (- x) \times 2 x + (- 6) \times 2 x + x^{2} \times (- 1) + (- x) \times (- 1) + (- 6) \times (- 1)$

Simplify.

$2 x^{3} - 2 x^{2} - 12 x - x^{2} + x + 6$

Collect 'like' terms.

$2 x^{3} - 3 x^{2} - 11 x + 6$

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	x	+1
x	x²	x
+3	3x	3

	x²	+3x	+1
x	x³	3x²	x
+2	2x²	6x	2

Expanding Multiple Brackets (AQA GCSE Maths)

Revision Note

How do I expand two brackets using FOIL?

How do I expand two brackets using a grid?

How do I expand a bracket squared?

How do I expand three brackets?

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Author: Mark