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) \(~2(2x^2-y^2-4x^2y^2)~\)

\(4x^2-2y^2-8x^2y^2\)

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