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BASIC ALGEBRA II
Grade 10 CAPS Mathematics
Video Series
Outcomes for this Video
In this DVD you will:
• Revise factorization.
LESSON 1.
• Revise simplification of algebraic fractions.
LESSON 2.
• Discuss when trinomials can be factorized.
LESSON 3.
2
In this Video we will focus on:
• Single Term Algebraic Fractions and their
Simplification (Lesson 1)
• Working with combinations of Algebraic
Fractions (Lesson 2)
• Simplifying Compound Algebraic
Fractions (Lesson 3)
Single Term
Algebraic
Fractions and their
Simplification Grade 10
CAPS Mathematics Video Series
Lesson 1
Outcomes for this Lesson
In this lesson we will:
• Revise basic knowledge on Algebraic
Fractions
• Simplify single term Algebraic
Fractions
The following information about the use of brackets is
helpful when working with algebraic expressions:
1.) x y y x
2.) x y x y
3.) x y y x
4.) x y x y
5.) ; is evenn n
x y y x n
6.) ; is unevenn n
y x x y n
Some Algebra Basics: The use of brackets
Fraction has the form a
b
and are integersa b
is never zero!!!!b
Examples of number fractions:
1 3 11 17; ; ;
2 4 3 213
Any integer number is a number fraction!!
3 10 10 10 13 ; 10 ; 1=
1 1 1 1 1
All the number fractions form the set of rational numbers denoted by
Numerator
Denominator
What is a Number Fraction (or Rational Number)?
An Algebraic Fraction also has the form a
b
and are expressions representing real numbersa b
Examples :
2 3 2 2 1 3 3
2 1 2 1 1
20 2 5 4
35 7 5 7
x yt x y t xt
xy s x y s ys
First write numerical coefficients of an AF in index form:
2 3
5
2 4 2; ;
3 7
x y x tlv
t y xyx
Simplify by cancelling all common factors above and below
Numerator
Denominator never represents zero!b
GCD
What is an Algebraic Fraction (AF)?
• If the denominator of an algebraic fraction is zero, the expression is meaningless.
• No factor in the denominator can thus be zero.
• Hence the variable(s) in the denominator cannot have certain values.
1 41)
3 5x x
: 3 5x x Restrictions and
2
2
3 ( 3)2)
4 ( 2)( 2)
x x x x
x x x
: 2 2x x Restrictions and
Examples :
Restrictions linked to Expressions with
Algebraic Fractions
• Factorize both denominator and numerator
• Divide both by the same factor (s) or cancel
common factors
• Only non-zero factors can be cancelled!!
2 225
2 10
a b
a b
5 5
2 5
a b a b
a b
5
2
a b
5
2
a b
Factorise both numerator
and denominator.
Cancel common factor (s).
Preferred format.
Example 1
Simplification of Single Term Algebraic Fractions
4 3 2
2
6
9 3
x x x
x x
Factorise both numerator
and denominator.
Cancel common factor and
write in preferred form.
2 3 2
3 3
x x x
x x
2
3
x x
Note that we assume that 3x
Example 2
Simplification of Single Term Algebraic Fractions
Use exponent rule = where neededn
n m
m
xx
x
2
ax b x ab
ax abx
Factorisation
Stage 1.
ax ab x b
ax x b
a x b x b
ax x b
Factorisation
Stage 2.
Factorisation
Final Stage.
1x b a
ax x b
1a
ax
Simplification.
Cancel common factors.
Example 3
Simplification of Single Term Algebraic Fractions
Simplify the following Algebraic Fractions:
3 4
2 2 4
3
2
3
2
81)
12
2 182)
3
43)
6
a bd
a b cd
x x
x x
t t
t t
PAUSE VIDEO
• Do Tutorial 1
• Then View Solutions
Tutorial 1: Simplification of Single Term Algebraic
Fractions
Simplify the following Algebraic Fractions:
3 4
2 2 4
81)
312
4a bd
a b c bcd
a
Tutorial 1: Suggested Solutions
3
2
2 2 243)
6 2
3
3
t t t t tt
tt t t
t
t
3 2
2
2 92 182)
3
2 3 32 3
3 3
x x x x xx
x x x
x x
xx x
Working with
combinations of
Algebraic Fractions
Grade 10 CAPS Mathematics
Video Series
Lesson 2
Outcomes for this Lesson
In this lesson we will focus on:
• Addition and Subtraction of Algebraic
Fractions
• Multiplication and Division of Algebraic
Fractions
• Factorize denominators where necessary.
• Determine LCM of the denominators.
• Write all the fractions as one fraction with
LCM as denominator.
• Simplify the numerator of the resultant
algebraic fraction (If required).
Addition & Subtraction of Algebraic Fractions
Strategy
a c a c
b b b
Examples :
2 2 2
2 4 2(3 ) 4( ) 6 4 2 (3 2 ) 6 4
3 3 3 3 3
x xy x xy x y y
xy x x y x y x y xy
2 2 2
3 2 3 2
2 2 2
y t y t
x x x
Same Denominator
a c ad cb
b d bd
Different Denominators
0b ( ,3 ) 3LCM xy x xy
3 32 4
32 4 6 4
3 3 3
xy xy
xy x y
xy x xy xy
or
Addition & Subtraction of Algebraic Fractions
, 0b d
2 1
2 6
x x
3 2 1
6
x x
3 6 1
6
x x
2 7
6
x
LCM(2,6) = 6
Write as single fraction with
LCM as denominator.
Simplify fraction
Lowest
Common
Multiple
Example :
Addition & Subtraction of Algebraic Fractions –
Basic Example
2 11
2
xx
2 1 2 1
2
x x
2 2 2 1
2
x x
3
2
Write as single fraction
with LCM as denominator.
Simplify algebraic fraction
Example : Note that 1 is seen
1as and as
1 1
xx
Addition & Subtraction of Algebraic Fractions -
Example
2 2
21
ab a b
a b a b b a
2
1ab a b
a b a b a b a b
2ab a a b b a b a b a b
a b a b
2 2 2 22ab a ab ab b a b
a b a b
2 22 2a b
a b a b
2 22 a b
a b a b
2
Factorize denominators
and determine LCM.
Write as single fraction
with LCM as denominator.
Simplify fraction
Example :
Addition & Subtraction of Algebraic Fractions -
Example
Examples :
3 2
2 4 1
2 2 2
2 2 1
1 2 2 1
2 1
1 2 1
4
2 1
4
2 1
xF
x x x x
x
x x x x
x x x
x
x x
x
x x
2( 1), ( 1) 2 ( 1)LCM x x x x x
Factorize
Simplify
2 2
2 2 2 2
4 5 1
2( 1) 2( 1)
( 5)( 1) ( 1)( 1)
2 2
5 1)
2( 1) 2( 5)
( 5) 1
4 8
( 5) 1
x xF
x x x
x x
x x x x
x x
x x
x x
x
x x
LCM
Simplify
( 5),( 1) ( 5)( 1)LCM x x x x
Examples where factorization play a role
Simplify the following expressions completely:
1)
1 12)
4 1 5
3 5 7 1 13)
4 6 1 3
a c b
bc ba ac
a a
a a
x x x
x x
PAUSE VIDEO
• Do Tutorial 2
• Then View Solutions
Tutorial 2: Addition and Subtraction of
Algebraic Fractions
Simplify the following expressions completely: 2 2 2
1) a c b
bc ba a
a
ac
c b
bc
Tutorial 2: Suggested Solutions
2 21 12)
4 1 5
5 1 4 1 13 4
20 1 20 1
a a a a a
a a a a
a a
a a
2
3 5 7 1 13)
4 6 1 3
3 3 5 1 2 7 1 4 1
=12 1
26 15
12 1
x x x x x x
x x
x x
x
x
x
x x
x x
Examples :
2 2 3
3 2
2 2 3
3 2
2 2 4
2 2 3
24 2
8
24 2
8
6
6
a b x xy
ax ab
a b x xy
ax ab
a b x y
a b x
xy
Format
a c a c
b d b d
Note:
a a a
b b b
First multiply the signs,
then the numbers
Always try the final fraction using
the basic exponent rules of Algebra
simplify
More General:
2 2 32 2 3 3 2 32
3 3 2 23
22 2 2
3 155 155 3
xy a b yxy a b y xy a bxa b
ab aby y a bab by a
Multiplication of Algebraic Fractions
2
2 2
3 2
4 2 5 3
ax a x x
x x x
3 2
2 2 2 1 3
a x x x
x x x x
Factorise all numerators
and denominators.
3 2
2 2 2 1 3
a x x x
x x x x
Cancel common factors.
2 2 1
ax
x x
Simplified result.
Example :
Multiplication of Algebraic Fractions -
Example
, , is never zero!!!!b c daa d a db
c b c b c
d
Multiply with the
reciprocal of the
denominator fraction
2 5 2 4 8
3 4 3 5 15
5 4 and are reciprocals
4 5
4 3 2 4 8 2 4 8 2 64
7 8 5 7 3 5 7 3 5 105
Examples :
Reciprocal
3 8 and are reciprocals
8 3
Division of Algebraic Fractions
2 2 2
2
12 32 8 16 4
8 8 8
x x x x x x
x x x x
4 8 8 4
8 4 4 8
x x x x x x
x x x x
Factorise all numerators
and denominators.
4 8 8 4
8 4 4 8
x x x x x x
x x x x
Cancel common factors.
8
8
x x Simplified result
Example :
Multiplication and Division of
Algebraic Fractions - Example
Division to multiplication.
2
2
5 52 2
4 6 5 4 4
a b aa a a
b a a ab b
Factorise all numerators
and denominators.
2 1 5 1
4 5 1 4 1
a a a ba
b a a b a
5 1 4 1
4 2 1 5 1
a a b aa
b a a a b
Division to multiplication
Cancel common factors.
1
2 1
a
b
Simplified result.
Example :
Multiplication and Division of
Algebraic Fractions - Example
Simplify the following expressions completely:
2 2
2 2 2 2
2 2
2 2
2
2 2 2 2
2 2
21)
2 2 4 4 4 22)
2 2
21 1 423)
2
2 2 4 4 4 24)
2 2
ab a b b
a b a b a
a ab b a ab b
a ab b a b b a
x y x xy y
x x xy
a ab b a ab b
a ab b a b b a
PAUSE VIDEO
• Do Tutorial 3
• Then View Solutions
Tutorial 3: Multiplication and Division of
Algebraic Fractions
Simplify the following expressions completely:
Tutorial 3: Suggested Solutions
2 2
221) 2
a b a bab
a
ab a b b
a b a ba b ab a b
2
2 2 2
2
2
2
2
2 2 4 4 4 22)
2 2
2 2 2 =
22
= 1
a a
a b a b a b a b
a b a
b b a ab b b a
a a a
b
b b b
Simplify the following expressions completely:
Tutorial 3: Suggested Solutions
2
2
2 2
2
7 621 2
2
3 7
2 7 6
3
21 1 423)
2 6
2
x y x xy y
x x xy
x y x yx y x
x xy
x y xy
x x y x y
y
x x y
Simplify the following expressions completely:
Tutorial 3: Suggested Solutions
2 2 2 2
2 2
2 2 2
2 2 2
2 2
2 2
2 2 4 4 4 24)
2 2
0
a b a b a b
a b a b a
a ab b a ab b
a a
b a b
b b
a
a b b a
b
a b a b
Simplification of
Compound
Algebraic Fractions
Grade 10 CAPS Mathematics
Video Series
Lesson 3
34
• Compound fraction is a fraction in which numerator and/or denominator contain fractions.
• “Inside Denominators “ are those denominators which form part of fractions in the denominator and/or numerator parts of the compound fraction.
• The idea is to eliminate all these “inside denominators” i.e. to change the compound fraction into an ordinary algebraic fraction.
Explaining the structure of
Compound Algebraic Fractions
Strategy:
Multiply numerator and denominator of the compound fraction with the LCM of all “inside denominators”. Example
2
k m
m kk m
m k
What is LCM of all “inside denominators”?
2
k m
m kk m
m
mk
mk
k
Multiply numerator and denominator with
this LCM of all denominators
2 2
2 22
k m
k mk m
2
k m k m k m
k mk m
Simplify fraction
Simplifying Compound Algebraic Fractions
1 1
12 3
1
x x
x x
What is LCM of all “inside denominators”?
11 1
1
12 3
1
x x
x x
x x
x x
Multiply numerator and denominator with this LCM
1
2 1 3 1 1
x x
x x x x
2 2
2 1 2 1
2 2 3 3 1
x x
x x x x x
Simplify fraction
Simplifying Compound Algebraic Fractions -
Example
Example
What is LCM of all “inside denominators”?
Multiply numerator and denominator with this LCM
11
11
2 11
xx
xx
11
11
2 11
1
1
xx
xx
x
x
2 2
2 2
1 1
2 2 1 1 2 3 2 3
x x x x x
x x x x x x
Simplify fraction
1 1 1
2 1 1 1
x x x
x x x
Example
Simplifying Compound Algebraic Fractions -
Example
Simplify the following expressions completely:
Tutorial 4: Simplification of Compound
Algebraic Fractions
2
Simplify the following compound fractions:
12
1) 1
4
112
22) 1 2 1
2 3 3
a
a
xx
xx
PAUSE VIDEO
• Do Tutorial 4
• Then View Solutions
Simplify the following expressions completely:
Tutorial 4: Suggested Solutions
2
2
2
2
2
2
12
14
2 12
1 4 1 2 1 2 1
12
1) 1
4
2
aa
aa
a
a
a aa a a
a a a a
Simplify the following expressions completely:
Tutorial 4: Suggested Solutions
1 2 12 11 2 32 3 33 32
1 1 1 12 2 2 3
2 2
2 1 2 1 2 1
2 1 3 2 1 3 2
112
22) 1 2 1
2 3 3
1
xx x xx
x x xx x
x x x x x
x x
x
x
x
x
x
REMEMBER!
•Consult text-books for additional examples.
•Attempt as many as possible other similar
examples on your own.
•Compare your methods with those that were
discussed in the Video.
•Repeat this procedure until you are confident.
•Do not forget:
Practice makes perfect!
End of Video on Basic Algebra Part II
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