Bracket Expansion and Factorisation

Slideshow 15MathematicsMr SasakiRoom 307

BRACKET EXPANSION AND FACTORISATION

To recall how to expand pairs of brackets for a quadratic

To be able to factorise quadratics in the form 2 + b + c

To be able to solve quadratics in the form 2 + b + c = 0.

OBJECTIVES

Today, we are dealing with a certain form of polynomial. Each has a special name.

DEFINITIONS

4 This is a “constant”. It doesn’t change. It’s also a monomial (one term).

4 + 3 This is “linear”.

42 + 3 - 2 This is a “quadratic”.

23 - 42 + 3 - 2 This is a “cubic”.4 + 23 - 42 + 3 - 2 This is a “quartic”.75 + 4 +23 - 42 + 3 - 2 This is a “quintic”.

To expand a pair of brackets representing a quadratic, we multiply each term inside each bracket by each term in the other bracket.Here are the combinations.

EXPANDING BRACKETS

(𝑎+𝑏)(𝑐+𝑑)¿𝑎𝑐+¿𝑎𝑑+¿𝑏𝑐+¿𝑏𝑑

Notice that ab and cd are not combinations.

Try the example below.

EXPANDING BRACKETS

ExampleExpand (2 – 1)(4 + 6).

(2 - 1)(4 + 6)=82 +12 -4 -6=82 + 8 - 6

Try the worksheet!

ANSWERS

𝑥2+4 𝑥+3𝑥2−2 𝑥−8𝑥2−10 𝑥+21

𝑥2+10 𝑥+254 𝑥2+10 𝑥+4

15 𝑥2−19 𝑥+64 𝑥2+9𝑥−924 𝑥 2+46 𝑥−1816 𝑥2+64 𝑥+64−2 𝑥2+7 𝑥−3¼ 𝑥2−4 𝑥+16

Placing a quadratic into a pair of brackets is called “factorisation”. This is the opposite of expanding brackets and more difficult to do.

FACTORISATION

Let’s try a linear expression.ExampleFactorise 9 – 6.

9 - 6What is the largest factor that divides into 9 and 6?

3=3( )3 - 2

The contents of the bracket is divided by the coefficient outside.

FACTORISATION

A quadratic is more difficult.ExampleFactorise + 5x + 6.

+ 5 + 6We need to think of two numbers which add together to make 5 and multiply to make 6.2 and 3

=( ) ( )𝑥 𝑥

The term has a coefficient of 1 because has a coefficient of 1.

Each bracket contains .2 and 3 are positive so we get + 2 and + 3.

+ 2 + 3If you are unsure it’s right, expand it out to check!

FACTORISATION

Let’s try another example.ExampleFactorise - 5x - 36.

We need to think of two numbers which add together to make -5 and multiply to make -36.

Hint: 9 – 4 is 5 and 9 x 4 is 36.-9 and

4 -9 + 4 = -5-9 x 4 = -36

=( )( )𝑥 𝑥- 9 + 4We will only look at quadratic expressions where the coefficient of is 1.

Try the worksheet!

ANSWERS

(𝑥+3)(𝑥+1)(𝑥+3)(𝑥+5)(𝑥+3)(𝑥+4 )(𝑥+2 )2(𝑥+7 )2

(𝑥−7 )(𝑥−1)(𝑥−5 )2(𝑥+7)(𝑥+9)(𝑥+1)(𝑥+8)(𝑥−8)(𝑥+2)(𝑥+20)(𝑥+4 )

SOLVING QUADRATIC EQUATIONS THROUGH FACTORISATION

We now know how to factorise quadratics. But how do we solve them for f() = 0?[f() means a function of .]ExampleSolve - 6x + 5 = 0.( )( ) = 0𝑥 𝑥- 5 - 1

This means that – 5 = 0 and – 1 = 0.So = 5 or = 1.

SOLVING QUADRATIC EQUATIONS THROUGH FACTORISATION

ExampleSolve + 18x + 72 = 0.( )( ) = 0𝑥 𝑥+ 6 +

12So .Try the last worksheet!

ANSWERS

or or

(𝑥+9)(𝑥 – 2)or

(𝑥 – 10)(𝑥 –1)or

𝑥+9𝑥+3−9−3𝑥−11𝑥 – 2112𝑥−9𝑥 – 3937