Introduction
• This chapter focuses on developing your
skills with Algebraic Fractions
• At its core, you must remember that
sums with Algebraic Fractions follow
the same rules as for numerical versions
• You will need to apply these alongside
general Algebraic manipulation
Algebraic Fractions
You need to be able to rewrite
Fractions in their ‘simplest form’
One way is to find common factors,
and divide the fraction by them. The
factors must be common to every
term.
Example Questions
Divide by the
common
Factor (4)
In the second example, you cannot just
‘cancel the x’s’ as they are not common
to all 4 terms.
If you Factorise, you can then divide
by the whole Numerator, along with
the equivalent part on the
Denominator
Factorise the
Denominator
Divide by the
common
Factor (x + 3)
16
20
4
5
x3
2x  6
x3
2( x  3)
1
2
Divide by the
common
Factor (4)
Factorise the
Denominator
Divide by the
common
Factor (x + 3)
1A
Algebraic Fractions
You need to be able to rewrite
Fractions in their ‘simplest form’
One way is to find common factors,
and divide the fraction by them. The
factors must be common to every
term.
Sometimes you may have ‘Fractions
within Fractions’. Find a common
multiple you can multiply to remove
these all together (in this case, 6)
Example Questions
Multiply the
Numerator and
Denominator by 6
1
2
1
3
x 1
x  32
3x  6
2x  4
Factorise
Multiply the
Numerator and
Denominator by 6
Factorise
3( x  2)
2( x  2)
Divide by
(x + 2)
Divide by
(x + 2)
3
2
1A
Algebraic Fractions
You need to be able to rewrite
Fractions in their ‘simplest form’
One way is to find common factors,
and divide the fraction by them. The
factors must be common to every
term.
Sometimes you will have to Factorise
both the Numerator and Denominator.
Example Questions
Factorise the
Numerator AND
Denominator
x2 1
x2  4 x  3
( x  1)( x  1)
( x  1)( x  3)
Divide by (x + 1)
Factorise the
Numerator AND
Denominator
Divide by (x + 1)
x 1
x3
1A
Algebraic Fractions
You need to be able to rewrite
Fractions in their ‘simplest form’
One way is to find common factors,
and divide the fraction by them. The
factors must be common to every
term.
Another Example of a Fraction within
a Fraction…
You will usually be told what ‘form’ to
leave your answer in…
Example Questions
Multiply the
Numerator and
Denominator by x
Factorise
x  1x
x 1
x2  1
x2  x
Multiply the
Numerator and
Denominator by x
Factorise
( x  1)( x  1)
x( x  1)
Divide by (x + 1)
Split the
Fraction up
Divide by (x + 1)
x 1
x
x 1

x x
1
1
x
1A
Algebraic Fractions
You need to be able to multiply
and divide Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When multiplying Fractions, you
multiply the Numerators together, and
the Denominators together…
Example Questions
a)
1 3
3


10
2 5
b)
a c
ac


b d
bd
c)
3
5

5

9
It is possible to simplify a sum before
you work it out. This will be vital on
harder Algebraic questions
15
45

1
3
3

5
1 
1
5
9
3

1
3
1B
Algebraic Fractions
You need to be able to multiply
and divide Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When multiplying Fractions, you
multiply the Numerators together, and
the Denominators together…
It is possible to simplify a sum before
you work it out. This will be vital on
harder Algebraic questions
Example Questions
d)
e)
1a
c
b a
1

c
b
x 1
3
 2
2
x 1
Factorise
1
x 1
3
1
2
( x  1)( x  1)

3
2( x  1)
Multiply
Numerator
and
Denominator
1B
Algebraic Fractions
You need to be able to multiply
and divide Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When multiplying Fractions, you
multiply the Numerators together, and
the Denominators together…
Example Questions
a)
5 1

6 3
5 3
15


6 1
6

5
2
It is possible to simplify a sum before
you work it out. This will be vital on
harder Algebraic questions
When dividing Fractions, remember
the rule, ‘Leave, Change and Flip’
 Leave the first Fraction, change
the sign to multiply, and flip the
second Fraction.
1B
Algebraic Fractions
You need to be able to multiply
and divide Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When multiplying Fractions, you
multiply the Numerators together, and
the Denominators together…
Example Questions
a a

b c
b)
1
a c

b 1a

c
b
It is possible to simplify a sum before
you work it out. This will be vital on
harder Algebraic questions
When dividing Fractions, remember
the rule, ‘Leave, Change and Flip’
 Leave the first Fraction, change
the sign to multiply, and flip the
second Fraction.
1B
Algebraic Fractions
You need to be able to multiply
and divide Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When multiplying Fractions, you
multiply the Numerators together, and
the Denominators together…
It is possible to simplify a sum before
you work it out. This will be vital on
harder Algebraic questions
When dividing Fractions, remember
the rule, ‘Leave, Change and Flip’
 Leave the first Fraction, change
the sign to multiply, and flip the
second Fraction.
Example Questions
x  2 3x  6
 2
c)
x  4 x  16
x  2 x 2  16

x  4 3x  6
x  2 1( x  4)( x  4)

1
1 x4
3( x  2)
Leave, Change
and Flip
Factorise
1

( x  4)
3
Multiply the
Numerators
and
Denominators
1B
Algebraic Fractions
You need to be able to add and
subtract Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When adding and subtracting
fractions, they must first have the
same Denominator. After that, you
just add/subtract the Numerators.
Example Questions
a)
Multiply all
by 4
Add the
Numerators
1 3

3 4
4 9

12 12

Multiply all
by 3
Add the
Numerators
13
12
1C
Algebraic Fractions
You need to be able to add and
subtract Algebraic Fractions
The rules for Algebraic versions are
the same as for numerical versions
When adding and subtracting
fractions, they must first have the
same Denominator. After that, you
just add/subtract the Numerators.
Example Questions
b)
a
b
x
a b

x 1
Combine as a
single
Fraction
a bx

x x
a  bx
x
Imagine ‘b’ as
a Fraction
Multiply all
by x
Combine as a
single
Fraction
1C
Algebraic Fractions
You need to be able to add and
subtract Algebraic Fractions
Example Questions
c)
The rules for Algebraic versions are
the same as for numerical versions
When adding and subtracting
fractions, they must first have the
same Denominator. After that, you
just add/subtract the Numerators.
Factorise so
you can
compare
Denominators
Multiply by
(x - 1)
Expand the
bracket, and
write as a single
Fraction
3
4x
 2
x  1 x 1
3
4x

x  1 ( x  1)( x  1)
3( x  1)
4x

( x  1)( x  1) ( x  1)( x  1)

Simplify the
Numerator

3x  3  4 x
( x  1)( x  1)
7x  3
( x  1)( x  1)
Factorise so
you can
compare
Denominators
Expand the
bracket, and
write as a single
Fraction
Simplify the
Numerator
1C
Algebraic Fractions
x2 + 5x - 2
You need to remember how
to divide using Algebraic long
division
x-3
x3 + 2x2 – 17x + 6
x3 – 3x2
We are now going to look at
some algebraic examples..
1) Divide
(x – 3)
x3
+
2x2
– 17x + 6 by
5x2 - 17x + 6
3 5x
2xby
Second,
Third,
First,
Divide
Divide
Divide
x-2x
byby
x
So the answer is x2 + 5x – 2,
and there is no remainder
 x2
 5x
-2
This means that (x – 3) is a
factor of the original equation
 We then subtract
‘x2We

(x – then
3) from
subtract
what
‘-2(x
we
‘5x(x
started
–– 3)
3) from
from
withwhat
what
we have left
5x2 - 15x
- 2x + 6
- 2x + 6
0
1D
Algebraic Fractions
You need to remember how
to divide using Algebraic
long division
Always include all different
powers of x, up to the
highest that you have…
Divide x3 – 3x – 2 by (x – 2)
x2 + 2x + 1
x-2
x3 + 0x2 – 3x - 2
x3 – 2x2
2x2 – 3x - 2
First, divide
Second,
Third,
divide
divide
xx3 2x
by
by2xxby x
2x2 – 4x
x – 2
 You must include ‘0x2’ in
the division…
= 12x
x2
 So our answer is ‘x2 + 2x +
1. This is commonly known as
the quotient
Then, work out 1(x
x2(x– –2)
2x(x
–
2) and
and
subtract
subtract
from
from
what
whathave
you
you left
started
have
leftwith
x – 2
0
1D
Algebraic Fractions
You need to remember how
to divide using Algebraic
long division
Sometimes you will have a
remainder, in which case the
expression you divided by is
not a factor of the original
equation…
Find the remainder when;
2x3 – 5x2 – 16x + 10 is
divided by (x – 4)
 So the remainder is -6.
2x2 + 3x - 4
x-4
2x3 - 5x2 – 16x + 10
2x3 – 8x2
3x2 – 16x + 10
3 by
2 by
Third,divide
First,
Second,
divide
divide
2x
-4x
3x
by
xx
x
= 2x
-4 2
= 3x
3x2 – 12x
-4x + 10
-4x + 16
-6
2(x– –
Then, work out 2x
-4(x
Then,
4)
andwork
subtract
out 3x(x
from–
4) andyou
what
subtract
started
have
left
from
with
what you have left
1D
Algebraic Fractions
You need to remember how to
divide using Algebraic Long
Division
But, how do we deal with the
remainder?
4
19 ÷ 5 = 3 5
5 divides into 19
3 whole times…
The ‘remainder’
is the numerator
The ‘divisor’ is
the denominator
Another way to think of this sum is  19 = (3 x 5) + 4
26 ÷ 3
2
= 83
3 divides into 26
8 whole times…
The ‘remainder’
is the numerator
The ‘divisor’ is
the denominator
Another way to think of this sum is  26 = (8 x 3) + 2
1D
Algebraic Fractions
You need to remember how to
divide using Algebraic Long
Division
2x2 + 3x - 4
x-4
2x3 - 5x2 – 16x + 10
2x3 – 8x2
We did this division earlier 
3x2 – 16x + 10
So the sum we have including the
remainder is:
Remainder
Divisor
2x3 - 5x2 – 16x + 10 ÷ (x – 4) = 2x2
3x2 – 12x
-4x + 10
-6
+ 3x - 4 +
x-4
= 2x2 + 3x - 4 -
-4x + 16
-6
6
x-4
1D
Algebraic Fractions
You need to remember how to
divide using Algebraic Long Division
x2 + 3x - 3
x-1
Write (x3 + 2x2 – 6x + 1) ÷ (x – 1) in
the form:
 Ax
2
x3 + 2x2 – 6x + 1
x3 – x2
3x2 – 6x + 1
 Bx  C   x  1  D
3x2 – 3x
-3x + 1
 Just do the division as normal…
2
x3  2 x2  6 x  1   x  1  x 2  3x  3 
x 1
x3  2 x2  6 x  1   x 2  3x  3  x  1 2
-3x + 3
Multiply both
sides by (x – 1)
-2
1D
Summary
• We have practised our skills involving
Algebraic Fractions
• We have followed the same rules which
we use for numerical fractions
• We have also learnt how to deal
properly with remainders in Algebraic
division