DEPARTMENT OF MATHEMATICS
Core 3
Topic 1 – Algebraic Fractions
1
Algebraic Fractions
Cancelling down
algebraic fractions
+ and algebraic fractions
x and ÷
algebraic fractions
Writing an improper
algebraic fraction in
mixed number form
Cancelling by factorising
top and bottom
Expressing 2 or more
algebraic fractions as a
single, simplified fraction
Multiplying 2 algebraic
fractions and then
simplifying
Cancelling by spotting
difference of 2 squares
As before, with
integer/non-fractional
terms
Dividing 2 algebraic
fractions and then
simplifying
2
Algebraic Fractions
The first topic of C3 involves studying algebraic fractions. Firstly, we will look
at cancelling down algebraic fractions (the process is incredibly similar to
cancelling down simple fractions at school).
Cancelling an algebraic fraction:
To cancel down an algebraic fraction, we look to find a common factor on the top
and bottom of the fraction.
Worked Examples:
a)
b)
Leave the following fractions in their simplest form:
𝑥+3
5𝑥+15
2𝑥−6
7𝑥−21
3
Let’s look at an example involving quadratics:
c)
3𝑥 2 −4𝑥
6𝑥 2 −5𝑥−4
4
And finally, a special one to look out for; the difference of 2 squares:
d)
e)
𝑥 2 −25
𝑥 2 +𝑥−30
𝑥−
1
𝑥
𝑥+1
5
Notes:
6
Adding and Subtracting Algebraic Fractions:
At GCSE, to add or subtract a fraction, you must make the denominators the
same.The same applies at A2! First, we factorise our fractions, then check to see
what we need to multiply by to make their denominators the same.
Worked Examples:
a)
2(3𝑥+2)
9𝑥 2 −4
−
Write the following as a single fraction in its simplest form:
2
Top Tip:
3𝑥+1
You should always check you
have fully factorised both
fractions before trying to make
the denominators the same!
You may find that something
will cancel, making life a bit
easier.
b)
𝑥+1
3𝑥 2 −3
−
1
3𝑥+1
7
Occasionally, we’ll be expected to deal with a problem that doesn’t solely involve
fractions – there could be an integer in the mix, for example:
c)
2
𝑥−8
𝑓(𝑥 ) = 1 − (𝑥+4) + (𝑥−2)(𝑥+4) , 𝑥 ∈ ℝ, 𝑥 ≠ −4, 𝑥 ≠ 2
𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓(𝑥 ) =
𝑥−3
𝑥−2
8
Notes:
9
Multiplying and Dividing Fractions:
These are skills that are also covered at GCSE. They are not readily asked at
A2, though you could be expected to perform the skill it within a question. We’ll
look at an example of each:
Worked Examples:
a)
𝑥+1
b)
𝑥+2
2
𝑥+4
×
÷
Write as a single fraction in its simplest form:
3
𝑥 2 −1
3𝑥+6
𝑥 2 −16
10
Rules:
*** ALWAYS FACTORISE FIRST! ***
Multiplying Algebraic Fractions: Multiply numerators, multiply denominators. cancel
where possible
Dividing Algebraic Fractions: Keep, Switch, Flip!
Notes:
11
Writing an improper algebraic fraction in mixed number form:
Algebraic fractions basically represent division. When we write down something
like:
𝑥 2 + 5𝑥 + 6
𝑥+3
We are actually asking:
‘What is 𝑥 2 + 5𝑥 + 6
÷
𝑥 + 3 ?’
This is an example of an improper (top-heavy) algebraic fraction, as the
numerator is of a higher degree (𝑥 2 ) than the denominator (𝑥).
We have seen improper fractions at GCSE, and we have also seen mixed numbers.
For example:
12
5
2
is an improper fraction, whereas 2 would be its equivalent mixed number.
5
We can also convert improper algebraic fractions into mixed number form, by
performing algebraic division using the reverse grid method from last year.
Worked Examples:
a)
Write the following in mixed number form:
𝑥 4 + 𝑥 3 + 𝑥 − 10
𝑥 2 + 2𝑥 − 3
12
b)
Given that
3𝑥 4 −2𝑥 3 −5𝑥 2 −4
𝑥 2 −4
= 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 +
find the values of a, b, c, d and e.
𝑑𝑥+𝑒
𝑥 2 −4
,
𝑥 ≠ ±2,
13
14
Notes:
15
Algebraic Fractions – Questions
Exercise 1:
16
Exercise 2:
17
Exercise 3:
18
Exercise 4:
19
Mixed Exercise:
20
Solutions to all exercises:
Exercise 1:
Exercise 2:
21
Exercise 3:
Exercise 4:
22
Mixed Exercise:
23