Algebra and Funtions

7/23/2019 Algebra and Funtions

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Mathematics Presentation # Algebra and Functions



Algebra and Functions

• This chapter focuses on basic manipulation of

• Factorisation of quadratic equations

• It also goes over rules of Surds and Indices

• It is essential that you understand this whole cas it links into most of the others!



You can simplify epression by collecting like terms"#

terms that are same$ for eample%

Simplifying expressions (like term

& and ' (red) $ *y+&y (

and +,-. (purple) are like

this algebric epression#

a) & +*y+, - +&y-.

&- +*y+&y+,-.

* +/y-*



Simplifying expressions (like term

0ere are some other eamples of simplification of

terms%a)1( a+b+c )-&a+*c

1a+1b+1c-&a+*c

1a-&a+1b+1c+*c

a+1b+2c

3pand the bracket first$ by multip

the term outside it by each term ins

then simplify the epression #

3pand the first bracket# 4ultiply

term in second bracket by every ter

the last bracket once$ then simplify

epression by adding like terms"#

b)*( a+ )+( a+b )(*+b )

*a+*+*a+ab+*b+

*a+*a+*++ab+*b

1a+&+ab+*b



Factorisation

a)

5ommon Factor

3

b) x

c) 4x

d) 3xy

e) 3x

Factorising is the opposite of epanding brackets# An epressi

into brackets by looking for common factors#

3 9 x + 3( 3) x= +2 5 x x− ( 5) x x= −

28 20 x x+ 4 (2 5) x x= +2 29 15 x y xy+ 3 (3 5 ) xy x y= +

23 9 x xy− 3 ( 3 ) x x y= −



Factorisation

3amples%

a)

The * numbers in brackets must%

4ultiply to give c"

Add to give b"

A 6uadratic 3quation has the form7

a * + b + c

8here a$ b and c are constantsand a 9 :#

You can also Factorise theseequations#

;3434<3;

An equation with an *" indoes not necessarily go into *brackets# You use * brackets whenthere are => 5ommon Factors"

26 8 x x+ +

( 2)( 4) x x= + +



Factorisation

3amples%

a)

The * numbers in bracket

must%

4ultiply to give c"

Add to give b"

A 6uadratic 3quationhas the form7

a * + b + c

8here a$ b and c areconstants and a 9 :#

You can also Factorisethese equations#

24 5 x x− −

( 5)( 1) x x= − +



Factorisation

3amples%

a)

The * numbers in brackets mus

4ultiply to give c"

Add to give b"


a * + b + c



(In this ca

This is known as the

difference of two squares"

* ' y* ? ( + y)( ' y)

225 x −

( 5)( 5) x x= + −



Factorisation

3amples%

a)

The * numbers in bracket

must%

4ultiply to give c"

Add to give b"


a * + b + c



25 45 x −

25( 9) x= −

5( 3)( 3) x x= + −



ndices

•

8e can solve epression involving indices

(powers) using specific set of rules%



ndices 3amples%

a)

b)

c)

d)

e)

f)

You need to be able to simplify epressions involving

Indices$ where appropriate#m n m n

a a a+× =

m n m na a a

−÷ =

( )m n mna a=

1m

ma a

−

=

1

mma a=

( )n

nmma a=

2 5 x x× 7 x=

2 32 3r r ×56r =

4 4b b÷ 0b= 1=

3 56 3 x x− −

÷ 2 x=

( )2

3 22a a×82a=

6 22a a×

( )3

2 43 x x÷6 427 x x÷

27 x=



ndices

3amples%

a)

b)

c)

d)

The rules of indices can also be applied to rational

numbers (numbers that can be written as a fraction)

m n m na a a

+× =m n m n

a a a−÷ =

( )m n mna a=

1mma

a

−

=

1

mma a=

( )n

nmma a=

4 3 x x−

÷ 7

x=

1 3

2 2 x x×4

2 x= =

23 3( ) x

23

3 x×

= =

1.5 0.252 4 x x−

÷

=



ndices 3amples%


numbers (numbers that can be written as a fraction) a)

b)

c)

d)

m n m na a a

+× =m n m n

a a a−÷ =

( )m n mna a=

1mma

a−

=

1

mma a=

( )n

nmma a=

1

29 9=3=

1

364 3 64=4=

3

249 ( ) 3

49=343=

3

225−

3

2

1

25

= =



ndices 3amples%


numbers (numbers that can be written as a fraction)a)

b)

m n m na a a

+× =m n m n

a a a−÷ =

( )m n mna a=

1mma

a−

=

1

mma a=

( )n

nmma a=

12

3

− ÷

12

3

= ÷

3

2=

1

31

8

÷

3

3

1

8

= ÷ ÷

1

2=



Surds

Surds are the irrational root of integers$ they are

form of numbers with infinite decimals# The ro

the prime numbers are surds#



Surds There are certain amount of rules for simplification$ addition$

subtraction$ multiplication$ division and rationali@ation of sur

•



Surds Simplification of Surds%

Breaking surds down (simplifying) is just a case of prime factorization.

a) can be simplified as #

b) can be simplified as

c) can be simplified as

This is equal to 2, since we alread

"ince we !now that 5#5($25, there%



Surds Addition and Subtraction in surds%

You can only add or subtract ‘like surds’ alt!oug! some surds !a"e to be s

get like surds and t!en t!ey can be eit!er added or subtracted. #or exampl

%!e abo"e two are examples of normal addition and subtraction in surds. &

more complicated ones$

's you can see in t!e abo"e example t!e roots first !ad to be simplified to g

terms’ and t!en t!ey were added.

•



Surds 4ultiplication and division in surds%

e !a"e already studied t!e properties of multiplying and di"iding two surd

some examples as well$

•

'ccording to t!e first multiplication rule of sur

s*uare root of + multiplied by s*uare root of +

%!is *uestion is related to t!e second m

rule according to w!ic!

%!is answer is obtained by using t!e di"ision



Surds 3panding brackets with surds%

Brackets wit! surds are expanded just like normal brackets except for t!ismultiplying surds instead of normal integers. You may be expected to expa

double brackets. -ost of t!ese *uestions re*uire full workings #or xamp

(a)

(b)

•

%!e s*uare root of / is multiplied by e"ery term

bracket since t!ere are no like terms left t!e e*u

furt!er simplified.

%!e terms in t!e first bracket are first multiplied

in t!e second bracket once t!en t!e like terms a

t!e e*uation is furt!er simplified.



Surds ;ationali@ing the denominator%

-at!ematicians !ate !a"ing an irrational denominator. ' surd is an irrati

You will come across two types of fractions w!ere you will !a"e to rationa

denominator.

0) t!e first one being a fraction wit! single surd "alue . %!is can be ration

by multiplying bot! t!e numerator and t!e denominator by t!e surd numbe

•



Surds ;ationali@ing the denominator%

/) %!e second type of fraction will be w!en t!e denominator !as / "alues

addition or subtraction sign$

1n t!is example t!e fraction is irrational t!erefore to make it rational we m

t!e numerator and t!e denominator wit! .

•

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Algebra and Funtions