Expanding Brackets (Cambridge O Level Maths)

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Mark

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12 April 2024

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Expanding One Bracket

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 $3 x \times x + 3 x \times 2$ which simplifies to $3 x^{2} + 6 x$

Beware of minus signs

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

Worked example

(a)

Expand

4 x (2 x - 3)

Multiply the $4 x$ term outside the brackets by both terms inside the brackets, watch out for negatives!

$4 x \times 2 x + 4 x \times (- 3)$

Simplify.

$8 x^{2} - 12 x$

(b)

Expand

- 7 x (4 - 5 x)

Multiply the $- 7 x$ outside the brackets by both terms inside the brackets, watch out for negatives!

$(- 7 x) \times 4 + (- 7 x) \times (- 5 x)$

Simplify.

$- 28 x + 35 x^{2}$

Expand & Simplify

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. $4 (x + 7) + 5 x (3 - x)$
- 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 $4 \times x + 4 \times 7$ , and simplifies to $4 x + 28$ , the second set of brackets expands to $5 x \times 3 + 5 x \times (- x)$ and simplifies to $15 x - 5 x^{2}$
- So, $4 (x + 7) + 5 x (3 - x) = 4 x + 28 + 15 x - 5 x^{2}$
STEP 2: Collect together like terms
- E.g. $4 (x + 7) + 5 x (3 - x) = 19 x + 28 - 5 x^{2}$

Worked example

(a)

Expand and simplify

2 (x + 5) + 3 x (x - 8)

Expand each set of brackets separately by multiplying the term outside the brackets by each of the terms inside the brackets.
Keep negative terms inside brackets so that you don't miss them!

$2 \times x + 2 \times 5 + 3 x \times x + 3 x \times (- 8)$

Simplify.

$2 x + 10 + 3 x^{2} - 24 x$

Collect 'like' terms.

$- 22 x + 10 + 3 x^{2}$

(b)

Expand and simplify

3 x (x + 2) - 7 (x - 6)

Expand each set of brackets separately by multiplying the term outside the brackets by each of the terms inside the brackets.
Keep negative terms inside brackets so that you don't miss them!

$3 x \times x + 3 x \times 2 + (- 7) \times x + (- 7) \times (- 6)$

Simplify.

$3 x^{2} + 6 x - 7 x + 42$

Collect 'like' terms.

$3 x^{2} - x + 42$

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Expanding Two 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 Three 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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Previous:Algebraic Roots & IndicesNext:Factorising

	x	+1
x	x²	x
+3	3x	3

	x²	+3x	+1
x	x³	3x²	x
+2	2x²	6x	2

Expanding Brackets (Cambridge O Level Maths)

Revision Note

How do I expand a bracket?

Beware of minus signs

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

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