BASIC ALGEBRA II resources... · 14 1) xx35 Restrictions and ... •Determine LCM of the denominators. •Write all the fractions as one fraction with LCM as denominator. •Simplify

$Page 1: BASIC ALGEBRA II resources... · 14 1) xx35 Restrictions and ... •Determine LCM of the denominators. •Write all the fractions as one fraction with LCM as denominator. •Simplify$
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


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


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


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


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


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


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


and denominators.

4 8 8 4

8 4 4 8

x x x x x x

x x x x


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


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


1

2 1

a

b

Simplified result.

Example :

Multiplication and Division of

Algebraic Fractions - Example


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


Tutorial 3: Multiplication and Division of

Algebraic Fractions



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



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



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


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


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


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


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




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



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

Documents

BASIC ALGEBRA II resources... · 14 1) xx35 Restrictions and ... •Determine LCM of the denominators. •Write all the fractions as one fraction with LCM as denominator. •Simplify