Transcript “Teach A Level Maths” Vol. 2: A2 Core Modules 35: Algebraic Fractions © Christine Crisp Algebraic Fractions We need to be able to • add and subtract.

“Teach A Level Maths”
Vol. 2: A2 Core Modules
35: Algebraic Fractions
© Christine Crisp
Algebraic Fractions
We need to be able to
•
add and subtract fractions
•
multiply and divide fractions
Before we look at algebraic fractions we’ll have a
reminder of arithmetic fractions.
Algebraic Fractions
Adding and subtracting fractions.
e.g. 1.
2
4

3
7
We need the denominators to be the same.
The smallest number both denominators divide into
is 21.
21 is the lowest common denominator.
2
4
2 7  4 3


3
7
3 7
st denominator26
The 2
has been multiplied by 73, so we
1nd
 the numerator by 3.
multiply 21
7
So,
Algebraic Fractions
e.g. 2.
2
4

3
9
Since 3 is a factor of 9, the lowest common
denominator is 9 not 27.
So,
2
4

 2 3  4
3
9
9
10
The 2nd denominator
unchanged.
The 1st denominator
by 3, so we
 has been ismultiplied
9 numerator by 3.
multiply the
Algebraic Fractions
e.g. 3.
2
4

21
15
3 is now a factor of both denominators.
We’ll write the factors out to see what we’ve got.
So,
2
4
2
4



21
15
3 7 3 5
The lowest common denominator is the smallest
number that both denominators divide into.
So, we need 3  7  5 ( 105)
So,
2
4

 2 5  4 7
3 7
3 5
3 7 5
nd denominator
The
so we
we
The 12st
denominator has
has been
been18multiplied
multiplied by
by 57, so
 
5.
multiply the numerator
by 7.
105
Algebraic Fractions
We use the same ideas when adding and
subtracting algebraic fractions.
I’ll put a similar arithmetic fraction beside each
example.
Algebraic Fractions
Arith.
2
4

3
7
2 7  4 3

3 7
e.g. 1.
2
4

x 1
x2
The denominators don’t share any factors
2( x  2)  4( x  1)

( x  1)( x  2)
14
 1212stnd denominator
4x  4
2 xbeen
 4 multiplied
The
The
denominatorhas
has
been
multiplied
by
by (x
(x2)
1),,

so we
we multiply
multiply the
the numerator
(x2)
1)..
(numerator
x  1)( x  2by
)by (x
3  7 so
6x
26


( x  1)( x  2)
21
The difference here is that we don’t
multiply the factors together.
Algebraic Fractions
Arith.
e.g. 2.
2
4x

x 1
( x  1) 2
2
4

st denominator is a factor of the 2nd
The
1
3
9
2( x  1)  4 x
2 3  4


2
(
x

1
)
9
nd denominator is unchanged.
st denominator
The
2
The
1
has
been
multiplied by (x  1),
2
x

2

4
x
64


2
so
we
multiply
the
numerator
by (x  1).
(
x

1
)
9
10

9
6x  2

( x  1)( x  2)
We factorise the numerator if possible
2( 3 x  1)

( x  1)( x  2)
Arith.
2
4

21
15
2
4


3 7 3 5
Algebraic Fractions
e.g. 3.
2
x  4x  3
2

3
x2  3x  2
We have to find the factors
2
3


( x  1)( x  3)
( x  1)( x  2)
The denominators share a factor
Arith.
2
4

21
15
2
4


3 7 3 5
Algebraic Fractions
e.g. 3.
2
x  4x  3
2

3
x2  3x  2
We have to find the factors
2
3


( x  1)( x  3)
( x  1)( x  2)
The denominators share a factor
2 5  4 7
2( x  2)  3( x  3)


3 7 5
( x  1)( x  3)( x  2)
10  28
 3 x  9 by
x  4multiplied
ndstdenominator
The
The21
denominatorhas
has2been
been
multiplied
by(x(x3)2),,

3 7 5
xnumerator
 1)( x  3)(
x (x(x
2)3)2). .
so
sowe
wemultiply
multiplythe
the(numerator
by
by
18
 x5
 

105
( x  1)( x  3)( x  2)
Algebraic Fractions
And one more . . .
e.g. 4.
2x  1
x  x2
2

x3
x2  x  6
2x  1
x3


( x  1)( x  2)
( x  2)( x  3)
x2  x  6

( x  1)( x  2)( x  3)
( 2 x  1)( x  3)  ( x  3)( x  1)

( x  1)( x  2)( x  3)
( x  3)( x  2)

( x  1)( x  2)( x  3)
2
2
2
x

5
x

3

(
x
 4 x  3)

( x  1)( x  2)( x  3)
2
2
2
x

5
x

3

x
 4x  3

( x  1)( x  2)( x  3)
1
1

x 1
Algebraic Fractions
SUMMARY
 To simplify a sum or difference of algebraic
fractions,
•
factorise any quadratic terms,
•
find the lowest common denominator, taking
care not to repeat factors ( unless they are
repeated in one of the fractions ),
multiply out the numerator and collect like
terms,
•
•
factorise the numerator.
Algebraic Fractions
Exercise
Express each of the following as a single fraction in
its simplest form.
1.
3.
4.
2
5

x3 x4
2.
2x  1
3x  2

3
2x  3

x  3 ( x  2)( x  3)
x  5x  6 x  x  6
x3
4

x2  1 x2  x
2
2
Algebraic Fractions
Solutions:
1.
2
5

x3 x4
2( x  4)  5( x  3)

( x  3)( x  4)
2 x  8  5 x  15

( x  3)( x  4)
7x  7

( x  3)( x  4)
7( x  1)

( x  3)( x  4)
Solutions:
2.
Algebraic Fractions
3
2x  3

x  3 ( x  2)( x  3)
3( x  2)  ( 2 x  3)

( x  3)( x  2)
3x  6  2x  3

( x  3)( x  2)
1 x3

( x  3)( x  2)
1

x2
Algebraic Fractions
Solutions:
2x  1
3.

3x  2
x  5x  6 x2  x  6
2x  1
3x  2


( x  2)( x  3) ( x  2)( x  3)
2
( 2 x  1)( x  2)  ( 3 x  2)( x  2)

( x  2)( x  3)( x  2)
2x2  3x  2  3x2  4x  4

( x  2)( x  3)( x  2)
5x2  x  6

( x  2)( x  3)( x  2)
(5 x  6)( x  1)

( x  2)( x  3)( x  2)
Solutions:
4.
Algebraic Fractions
x3

4
x 1 x  x
x3
4


( x  1)( x  1) x( x  1)
2
2
x( x  3)  4( x  1)

x( x  1)( x  1)
2
x  3x  4x  4

x( x  1)( x  1)
2
x  x4

x( x  1)( x  1)
Algebraic Fractions
 Multiplying and dividing fractions is easier than
adding or subtracting them.
Method:
•
For division, turn the 2nd fraction upside down
and multiply.
•
Factorise any parts that will factorise.
•
Cancel factors.
Algebraic Fractions
2
2
x

x

2
x
 x2
e.g. 1. Simplify
 2
2
x 4
x  3x  2
2
2
x

x

2
x
 x2
Solution:
 2
2
x 4
x  3x  2
( x  1)( x  2)
( x  1)( x  2)


( x  2)( x  2)
( x  1)( x  2)
x 1

x2
Algebraic Fractions
e.g. 2. Simplify
x2  4
x  x6
x2  4
Solution:
x  x6
x2  4
2

x  x6
2
2



x2  5x  6
x2  9
x2  5x  6
x2  9
x2  9
x2  5x  6
( x  2)( x  2)
( x  3)( x  3)


( x  2)( x  3)
( x  2)( x  3)
x2

x2
Algebraic Fractions
Exercise
Simplify the following:
1.
2.
4x  8
x  7 x  12
2

x2  5x  6
x  10 x  24
2
x 2  16
x2  2x  8

x2  x  2
x2  4x
Algebraic Fractions
Solutions:
1.
4x  8
x  7 x  12
2

x 2  16
x2  2x  8
4( x  2)
( x  4)( x  4)


( x  3)( x  4) ( x  4)( x  2)
4

( x  3)
Algebraic Fractions
x2  5x  6
2.
x  10 x  24
2

x  5x  6
2
x  10 x  24
2


x2  x  2
x2  4x
x  4x
2
x  x2
2
( x  1)( x  6)
x ( x  4)


( x  4)( x  6) ( x  1)( x  2)
x

x2
Algebraic Fractions
Solving equations containing algebraic fractions
e.g. 1 Solve the following to find the value of x
5
2
2


x4 x3 x
Solution:
We just multiply the whole equation by the lowest
common denominator of BOTH sides of the equation.
x( x  4)( x  3) :
5 xI’ve
( x put
4)( xthe
 3x) first
2 x(because
x  4)( xitmakes
3) 2ax(later
x  4stage
)( x  3)


easier.
x4
x3
x
So, multiply by
The denominators have all cancelled so we just have
a quadratic equation to solve.
Algebraic Fractions
5 x( x  4)( x  3) 2 x( x  4)( x  3) 2 x( x  4)( x  3)


x4
x3
x
So,
5 x( x  3)  2 x( x  4)  2( x  4)( x  3)

5 x 2  15 x  2 x 2  8 x  2 x 2  2 x  24

x  25 x  24  0

( x  1)( x  24)  0
2

x  1 or
x  24
Algebraic Fractions
Exercise
Solve the following equations to find the value of x:
1.
4
4

1
x 1 x 1
3.
2.
1
1
2


x 1
x x2
2
3

x  2 x 1
Algebraic Fractions
1.
4
4

1
x 1 x 1
Solution: Multiply by ( x  1)( x  1) :
4( x  1)( x  1) 4( x  1)( x  1)

 1( x  1)( x  1)
x 1
x 1
4( x  1)  4( x  1)  1( x  1)( x  1)
2
4x  4  4x  4  x  1
9  x2
Answer: x  3 or  3
Algebraic Fractions
2.
2
3

x  2 x 1
Solution: Multiply by ( x  2)( x  1) :
2( x  2)( x  1) 3( x  2)( x  1)

x2
x 1
2( x  1)  3( x  2)
2x  2  3x  6
8  x
Answer:
x  8
Algebraic Fractions
3.
1
1
2


x 1
x x2
x( x  1)( x  2) :
x( x  1)( x  2) x( x  1)( x  2) 2 x( x  1)( x  2)


x 1
x
x2
Solution: Multiply by
x( x  2)  ( x  x  2)  2 x( x  1)
2
x 2  2x  x 2  x  2  2x 2  2x
0  2x  3x  2
0  ( x  2)( 2 x  1)
2
Answer:
x  2 or x   12
Algebraic Fractions
Algebraic Fractions
The following slides contain repeats of
information on earlier slides, shown without
colour, so that they can be printed and
photocopied.
For most purposes the slides can be printed
as “Handouts” with up to 6 slides per sheet.
Algebraic Fractions
SUMMARY
 To simplify a sum or difference of algebraic
fractions,
•
factorise any quadratic terms,
•
find the lowest common denominator, taking
care not to repeat factors ( unless they are
repeated in one of the fractions ),
multiply out the numerator and collect like
terms,
•
•
factorise the numerator.
Algebraic Fractions
Arith.
2
4

3
7
2 7  4 3

3 7
14  12

3 7
26

21
e.g. 1.
2
4

x 1
x2
The denominators don’t share any factors
2( x  2)  4( x  1)

( x  1)( x  2)
2x  4  4x  4

( x  1)( x  2)
6x

( x  1)( x  2)
The difference here is that we don’t
multiply the factors together.
Algebraic Fractions
Arith.
e.g. 2.
2
4x

x 1
( x  1) 2
2
4

st denominator is a factor of the 2nd
The
1
3
9
2( x  1)  4 x
2 3  4


2
(
x

1
)
9
64

9
10

9
2x  2  4
( x  1) 2
6x  2

( x  1)( x  2)

We factorise the numerator if possible
2( 3 x  1)

( x  1)( x  2)
Arith.
2
4

21
15
2
4


3 7 3 5
2 5  4 7

3 7 5
10  28

3 7 5
18
 
105
Algebraic Fractions
e.g. 3.
2
x  4x  3
2

3
x2  3x  2
We have to find the factors
2
3


( x  1)( x  3) ( x  1)( x  2)
The denominators share a factor
2( x  2)  3( x  3)

( x  1) ( x  3)( x  2)
 2x  4  3x  9
( x  1)( x  3)( x  2)
 x5

( x  1)( x  3)( x  2)
Algebraic Fractions
2x  1
e.g. 4.
x  5x  6
2

x 1
x2  x  6
2x  1
x 1


( x  2)( x  3)
( x  2)( x  3)
( 2 x  1)( x  2)  ( x  1)( x  2)

( x  2)( x  3)( x  2)
2
2
2
x

5
x

2

(
x
 x  2)

( x  2)( x  3)( x  2)
2
x
 6x  4
2
x

5
x

2

x

x

2


( x  2)( x  3)( x  2)
( x  2)( x  3)( x  2)
2
2
Algebraic Fractions
 Multiplying and dividing fractions is easier than
adding or subtracting them.
Method:
•
For division, turn the 2nd fraction upside down
and multiply.
•
Factorise any parts that will factorise.
•
Cancel factors.
Algebraic Fractions
2
2
x

x

2
x
 x2
e.g. 1. Simplify
 2
2
x 4
x  3x  2
2
2
x

x

2
x
 x2
Solution:
 2
2
x 4
x  3x  2
( x  1)( x  2)
( x  1)( x  2)


( x  2)( x  2)
( x  1)( x  2)
x 1

x2
Algebraic Fractions
e.g. 2. Simplify
x2  4
x  x6
x2  4
Solution:
x  x6
x2  4
2

x  x6
2
2



x2  5x  6
x2  9
x2  5x  6
x2  9
x2  9
x2  5x  6
( x  2)( x  2)
( x  3)( x  3)


( x  2)( x  3)
( x  2)( x  3)
x2

x2