Expanding Single Brackets

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.

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