slideshow 15 mathematics mr sasaki room 307 bracket expansion and factorisation
TRANSCRIPT
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 - 6
What is the largest factor that divides into 9 and 6?
3=3( )3 - 2
The contents of the bracket is divided by the coeffi cient outside.
FACTORISATION
A quadratic is more diffi cult.
ExampleFactorise + 5x + 6.
+ 5 + 6
We 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 coeffi cient of 1 because has a coeffi cient 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 coeffi cient 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
Example
Solve + 18x + 72 = 0.( )( ) = 0𝑥 𝑥+ 6 +
12So .
Try the last worksheet!