# Unit 8 Section 1 : Expansion of single brackets

In this section we consider how to expand (multiply out) brackets to give two or more terms, as shown below:
 3 ( x + 6 ) = 3 x + 18
We will start by revising some negative number operations, then move on to multiplying out the brackets.

### Negative number operations

When you expand brackets, you often need to multiply a mixture of positive and negative items.

If you are multiplying two items with the same sign together, the answer is positive.
For example, +2 × +5 = +10 and -2 × -5 = +10

If you are multiplying two items with different signs together, the answer is negative.
For example, +2 × -5 = -10 and -2 × +5 = -10

Practice Questions
Work out the answer to each of these questions then click on the button marked to see whether you are correct.
(a) What is -7 × +4 ?

(b) What is -3 × -6 ?

(c) What is -4 × 9x ?

(d) What is -3(9 - 13) ?    [HINT: -3(9 - 13) means -3 × (9 - 13)]

### Expanding brackets

Look at the expression below:
 -3 ( x - 6 )
To expand the brackets, you need to multiply the part outside the brackets by every part inside the brackets.
In this case you need to multiply -3 by both x and -6. Then we combine the results:

Practice Questions
Work out the answer to each of these questions then click on the button marked to see whether you are correct.
(a) Expand the brackets in: -4(x - 3)

(b) Expand the brackets in: -3(9 - x)

(c) Expand the brackets in: 4(2x - 12)

### Expanding more complicated brackets

Here is a more complicated expression with brackets to expand:
 -5x ( 2x + 6y )
In this case you need to multiply -5x by both 2x and 6y.
 Working: -5x × 2x = -5 × 2 × x × x = -10 × x² = -10x² -5x × 6y = -5 × 6 × x × y = -30 × xy = -30xy
Then we combine the results:

Practice Questions
Work out the answer to each of these questions then click on the button marked to see whether you are correct.
(a) Expand the brackets in: 2x(x + 5)

(b) Expand the brackets in: -4x(x - y)

(c) Expand the brackets in: -7x(3y - 2x)

## Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.

Question 1
Work out the answers to the following questions involving negative numbers.
 (a) -11 × -4 (b) -6 × 4 (c) 8 × -7 (d) 8x × 4 (e) 6 × (8 - 10) (f) 5 × (3 - 10) (g) 7(11 - 4) (h) -4(6 - 17)
Question 2
Complete the multiplication tables below, and use your answers to expand the brackets below each table.
(a)
 × x 2 4
4(x + 2) =
(b)
 × x -7 5
5(x - 7) =
(c)
 × x 3 4
4(x + 3) =
(d)
 × 2x 5 5
5(2x + 5) =
Question 3
Expand the brackets in the questions below. There should be no brackets in your answers.
 (a) 4(x + 6) = (b) 3(x - 4) = (c) 5(2x + 6) = (d) 7(3x - 4) = (e) 3(2x + 4) = (f) 8(3x - 9) = (g) (-2)(x - 4) = (h) (-3)(8 - 2x) = (i) 5(3x - 4) = (j) 9(2x + 8) =
Question 4
Fill in the tables to help you expand the brackets in the given expressions below:
(a)
 × x -2 x x 2x 2x² x² 2 2x -2 -2x -x -x²
(b)
 × x -3 4 4x 4 4+x x -3x -4 12 -12 -12x
Question 5
Fill in the missing terms in the expansions below:
 (a) 4x(x + 8) = 4x² + (b) (-3)(2x - 7) = + 21 (c) 4x(x - 9) = 4x² - (d) 6x(x - 7) = 6x² - (e) 3x(x - y) = 3x² - (f) (-4x)(2x + 8) = -8x² -
Question 6
Expand the brackets in the following expressions:
Question 7
 In each of the questions below, write an expression for the area of the rectangle using brackets, then expand the brackets. Your answers should not include any spaces or multiplication signs (like * or ×). An example of how to answer a question is shown on the right.

(a)
2
x + 4
Expression for the area of the rectangle using brackets:
(b)
12
x - 5
Expression for the area of the rectangle using brackets:
(c)
2x
5 + x
Expression for the area of the rectangle using brackets:
(d)
2x
x + 9
Expression for the area of the rectangle using brackets:
(e)
2x
3x - 2
Expression for the area of the rectangle using brackets: