# Lesson 8: Expanding a single bracket

Introduction

In this lesson, we're going to be writing expressions like $$~3(x+7)~$$ in the form $$~3x+21~$$. These two expressions are equivalent, they just look different. Sometimes the bracketed form is more useful to us and sometimes the non-bracketed form is what we need. Don't forget, the topic Algebraic expressions is all about changing the form of expressions.

SECTION A

The first thing you need to know is that $$~3(x+7)~$$ means $$~3\times (x+7)~$$ which means three lots of everything in the bracket. Think about this for a minute because if you're happy with this, you will easily understand the examples below.

Example 1

Expand $$~3(x+7)~$$

Three times everything in the bracket means $$~3\times x~$$ AND $$~3\times 7~$$

$$~3\times x = 3x~$$

$$~3\times 7 = 21~$$

Answer: $$~3x+21~$$

Pretty straight forward, right?

Example 2

Expand $$~5(2x-7)~$$

$$~5\times 2x = 10x~$$

$$~5\times -7 = -35~$$

Answer: $$~10x-35~$$

Obviously you need to be able to multiply terms together (Lessons 2-3) and, of course, your negative number skills have to be up to scratch!

Example 3

Expand $$~4(-3x+2)~$$

$$~4\times -3x = -12x~$$

$$~4\times 2 = 8~$$

Answer: $$~-12x+8~$$

Example 4

Expand $$~-2(4x+5)~$$

$$~-2\times 4x = -8x~$$

$$~-2\times 5 = -10~$$

Answer: $$~-8x-10~$$

Example 5

Expand $$~2x(6x+9)~$$

$$~2x\times 6x = 12x^2~$$

$$~2x\times 9 = 18x~$$

Answer: $$~12x^2+18x~$$

OK, let's get some practice!

# Practise to master

SECTION A

Expand these single brackets.

01) $$~3(x+5)~$$

$$3x+15$$

02) $$~2(x+1)~$$

$$2x+2$$

03) $$~5(x-3)~$$

$$5x-15$$

04) $$~8(x-6)~$$

$$8x-48$$

05) $$~4(x+2)~$$

$$4x+8$$

06) $$~7(x-9)~$$

$$7x-63$$

07) $$~5(2x+3)~$$

$$10x+15$$

08) $$~2(5x+12)~$$

$$10x+24$$

09) $$~8(3x-5)~$$

$$24x-40$$

10) $$~3(9x-15)~$$

$$27x-45$$

11) $$~7(4x+7)~$$

$$28x+49$$

12) $$~6(6x-1)~$$

$$36x-6$$

13) $$~4(-3x+7)~$$

$$-12x+28$$

14) $$~7(-2x-5)~$$

$$-14x-35$$

15) $$~-2(5x+11)~$$

$$-10x-22$$

16) $$~-5(6x-1)~$$

$$-30x+5$$

17) $$~-3(3-4x)~$$

$$-9+12x$$

18) $$~-9(-x-10)~$$

$$9x+90$$

19) $$~7x(2x+2)~$$

$$14x^2+14x$$

20) $$~2x(10x+6)~$$

$$20x^2+12x$$

21) $$~4x(3x-5)~$$

$$12x^2-20x$$

22) $$~3x(4x-8)~$$

$$12x^2-24x$$

23) $$~5x(x+7)~$$

$$5x^2+35x$$

24) $$~x(x-1)~$$

$$x^2-x$$

25) $$~6x(3x+1)~$$

$$18x^2+6x$$

26) $$~3x(8x+13)~$$

$$24x^2+39x$$

27) $$~5x(4x-3)~$$

$$20x^2-15x$$

28) $$~9x(x-4)~$$

$$9x^2-36x$$

29) $$~4x(1-5x)~$$

$$4x-20x^2$$

30) $$~-2x(2x-7)~$$

$$-4x^2+14x$$

SECTION B

In this section, we're going to extend things a bit. Although, there are no new methods here so you could probably figure out how to do the practice questions yourself.

Example 1

Expand $$~2x^2(3xy+5y^2)~$$

$$~2x^2\times 3xy = 6x^3y~$$

$$~2x^2\times 5y^2 = 10x^2y^2~$$

Answer: $$~6x^3y+10x^2y^2~$$

Again, same method, easy if you're good at multiplying terms!

Example 2

Expand $$~3(2x^2-4x+1)~$$

$$~3\times 2x^2 = 6x^2~$$

$$~3\times -4x = -12x~$$

$$~3\times 1 = 3~$$

Answer: $$~6x^2-12x+3~$$

Three lots of everything in the bracket.

And some practice questions ...

# Practise to master

SECTION B

Expand these single brackets.

01) $$~y(2+5xy)~$$

$$2y+5xy^2$$

02) $$~y(7xy^2-1)~$$

$$7xy^3-y$$

03) $$~2a(4+5ab)~$$

$$8a+10a^2b$$

04) $$~3x(xy^2-3)~$$

$$3x^2y^2-9x$$

05) $$~y(7+5x^2y^2)~$$

$$7y+5x^2y^3$$

06) $$~q(p^2q-6)~$$

$$p^2q^2-6q$$

07) $$~2x(2y+7x^2)~$$

$$4xy+14x^3$$

08) $$~5x(2x^2-y^2)~$$

$$10x^3-5xy^2$$

09) $$~2xy(3y+2+4y^2)~$$

$$6xy^2+4xy+8xy^3$$

10) $$~3y(y+3x^2-2x^3)~$$

$$3y^2+9x^2y-6x^3y$$

11) $$~y(x^2y-4+12x)~$$

$$x^2y^2-4y+12xy$$

12) $$~2(4y-x^2-2x^2y^2)~$$

$$8y-2x^2-4x^2y^2$$

13) $$~2x(2xy+4+3y^2)~$$

$$4x^2y+8x+6xy^2$$

14) $$~3xy(3+2y-x)~$$

$$9xy+6xy^2-3x^2y$$

15) $$~x(12-4xy^2+y)~$$

$$12x-4x^2y^2+xy$$

16) $$~2(2x^2-y^2-4x^2y^2)~$$

$$4x^2-2y^2-8x^2y^2$$