1. Core 3
Algebraic Fractions

2. After completing this section you should be able to
‘cancel down’ algebraic fractions
Multiply together two or more algebraic fractions
Divide algebraic fractions
Add or subtract algebraic fractions
Convert an improper fraction into a mixed number

3. You treat algebraic fractions the same way as numerical ones
16 20
4 5
This cancels to
𝑥+3 2𝑥+6 = 𝑥+3 2(𝑥+3) = 1 2
𝑥+2 3𝑥+8
This is the simplest form, there are no common factors

4. How about 1 2 𝑥+1 1 3 𝑥+ 2 3 ? = 3𝑥+6 2𝑥+4 Method 1:
Look for a common factor of 2 and 3 i.e. 6
and then multiply the fraction by it
1 2 𝑥+1 × 𝑥 × 6
= 3𝑥+6 2𝑥+4
= 3(𝑥+2) 2(𝑥+2)
= 3 2

5. Step 1: factorise the numerator and denominator 𝑥+1 (𝑥 −1) 𝑥+1 (𝑥+3)
𝑥² −1 𝑥²+4𝑥+3
And now for . . .
Step 1: factorise the numerator and denominator
𝑥+1 (𝑥 −1) 𝑥+1 (𝑥+3)
Step 2: spot the common factor ( x + 1)
Step 3: cancel the common factor
Step 4: get the solution
(𝑥+1)(𝑥 −1) (𝑥+1)(𝑥+3)
= 𝑥 −1 𝑥+3

6. And finally 𝑥 − 1 𝑥 𝑥+1 Step 1:
𝑥 − 1 𝑥 𝑥+1
And finally
Step 1:
Remove the fraction by multiplying the numerator by x
𝑥² −1 𝑥(𝑥+1)
Step 2:
Factorise the numerator
(𝑥+1)(𝑥 −1) 𝑥(𝑥+1)
Step 3:
Cancel the common factor to get the answer
= 𝑥 −1 𝑥
(𝑥+1)(𝑥 −1) 𝑥(𝑥+1)

7. And now try exercise 1A page 3

8. To multiply fractions you multiply the numerators together and the denominators together (cancel down any common factors)
1 2 × 3 5 = 3 10
𝑎 𝑏 × 𝑐 𝑑 = 𝑎𝑐 𝑏𝑑
3 5 × 5 9
= × 5 3
= 1 3

9. Step 2: Cancel common factors
𝑥+1 2 × 3 𝑥² −1
And now
𝑥+1 2 × 3 (𝑥+1)(𝑥 −1)
Step 1: Factorise
Step 2: Cancel common factors
1 2 × 3 (𝑥−1)
3 2(𝑥−1)
Step 3: finish off

10. Dividing algebraic fractions
Same as it ever was
– invert the second fraction and multiply (cancel if you can)
𝑎 𝑏 ÷ 𝑎 𝑐 = 𝑎 𝑏 × 𝑐 𝑎 = 𝑐 𝑏
𝑥+2 𝑥+4 ÷ 3𝑥+6 𝑥² −16
Factorise and invert the second fraction then cancel
𝑥+2 𝑥+4 × 𝑥+4 𝑥 −4 3(𝑥+2) = 𝑥 −4 3

11. Your turn again Page 5 exercise 1B

12. Adding and Subtracting Fractions
All you need here is to have the same denominator
𝑎 𝑥 +𝑏
Lets look at
𝑎 𝑥 + 𝑏𝑥 𝑥
We can rewrite this as
𝑎+𝑏𝑥 𝑥
This now simplifies to

13. Multiply the first term by
3 𝑥+1 − 4𝑥 𝑥² −1
Here’s another one
3 𝑥+1 − 4𝑥 (𝑥+1)(𝑥 −1)
This can be written as:
Multiply the first term by
𝑥 −1 𝑥 −1
to put it over a common denominator
3(𝑥 −1) (𝑥+1)(𝑥 −1) − 4𝑥 (𝑥+1)(𝑥 −1)
Giving
3 𝑥 −1 −4𝑥 (𝑥+1)(𝑥 −1) = −𝑥 −3 (𝑥+1)(𝑥 −1)
Which simplifies to:

14. Dividing Algebraic Fractions
You can divide fractions if the power in the numerator is higher or equal to the power in the denominator. For example:
𝑥²+5𝑥+8 𝑥 −2
𝑥² −8+5 𝑥²+2𝑥+1
There are two methods – Long Division and the Remainder Theorem
This is too soul destroying to type in we will look at the examples on page 8 and 9 then do exercise 1D and 1E

15. And it’s your turn again Page 7 exercise 1C