Algebra and Funtions
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Transcript of Algebra and Funtions
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Mathematics Presentation # Algebra and Functions
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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!
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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-*
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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
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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= −
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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= + +
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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= − +
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Factorisation
3amples%
a)
The * numbers in brackets mus
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#
(In this ca
This is known as the
difference of two squares"
* ' y* ? ( + y)( ' y)
225 x −
( 5)( 5) x x= + −
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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#
25 45 x −
25( 9) x= −
5( 3)( 3) x x= + −
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ndices
•
8e can solve epression involving indices
(powers) using specific set of rules%
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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=
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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−
÷
=
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ndices 3amples%
The rules of indices can also be applied to rational
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
= =
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ndices 3amples%
The rules of indices can also be applied to rational
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=
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Surds
Surds are the irrational root of integers$ they are
form of numbers with infinite decimals# The ro
the prime numbers are surds#
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Surds There are certain amount of rules for simplification$ addition$
subtraction$ multiplication$ division and rationali@ation of sur
•
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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($25, there%
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Surds Addition and Subtraction in surds%
You can only add or subtract ‘like surds’ alt!oug! some surds !a"e to be s
get like surds and t!en t!ey can be eit!er added or subtracted. #or exampl
%!e abo"e two are examples of normal addition and subtraction in surds. &
more complicated ones$
'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.
•
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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
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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.
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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
•
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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! .
•