1. Algebraic Fractions

2. 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

3. Teachings for Exercise 1A

4. 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.
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
Example Questions
Divide by the common Factor (4)
Divide by the common Factor (4)
Factorise the Denominator
Factorise the Denominator
Divide by the common Factor (x + 3)
Divide by the common Factor (x + 3) 1A

5. 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
Multiply the Numerator and Denominator by 6
Factorise
Factorise
Divide by (x + 2)
Divide by (x + 2) 1A

6. 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
Factorise the Numerator AND Denominator
Divide by (x + 1)
Divide by (x + 1) 1A

7. 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
Multiply the Numerator and Denominator by x
Factorise
Factorise
Divide by (x + 1)
Divide by (x + 1)
Split the Fraction up 1A

8. Teachings for Exercise 1B

9. 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 a) b) c) 1B

10. 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 1 d) 1 e)
Factorise 1 1
Multiply Numerator and Denominator 1B

11. 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 a) 1B

12. 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 b) 1 1 1B

13. 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 c)
Leave, Change and Flip
Factorise 1 1 1 1
Multiply the Numerators and Denominators 1B

14. Teachings for Exercise 1C

15. 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
Multiply all by 3
Add the Numerators
Add the Numerators 1C

16. 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
Example Questions b)
Imagine ‘b’ as a Fraction
Multiply all by x
Combine as a single Fraction
Combine as a single Fraction 1C

17. Algebraic Fractions c) 1C
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 c)
Factorise so you can compare Denominators
Factorise so you can compare Denominators
Multiply by (x - 1)
Expand the bracket, and write as a single Fraction
Expand the bracket, and write as a single Fraction
Simplify the Numerator
Simplify the Numerator 1C

18. Teachings for Exercise 1D

19. You need to remember how to divide using Algebraic long division
Algebraic Fractions x2 + 5x - 2
You need to remember how to divide using Algebraic long division
We are now going to look at some algebraic examples..
1) Divide x3 + 2x2 – 17x + 6 by (x – 3)
So the answer is x2 + 5x – 2, and there is no remainder
This means that (x – 3) is a factor of the original equation
x - 3
x x2 – 17x + 6
x3 – 3x2
5x x + 6
5x x
Second, Divide 5x2 by x 5x
We then subtract ‘5x(x – 3) from what we have left
Third, Divide -2x by x -2
We then subtract ‘-2(x – 3) from what we have left
First, Divide x3 by x x2
We then subtract ‘x2(x – 3) from what we started with
- 2x + 6
- 2x + 6 1D

20. You need to remember how to divide using Algebraic long division
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)
 You must include ‘0x2’ in the division…
 So our answer is ‘x2 + 2x + 1. This is commonly known as the quotient x2 + 2x + 1
x - 2
x x2 – 3x - 2
x3 – 2x2
2x2 – 3x - 2
Second, divide 2x2 by x
= 2x
Then, work out 2x(x – 2) and subtract from what you have left
Third, divide x by x
= 1
Then, work out 1(x – 2) and subtract from what you have left
First, divide x3 by x
= x2
Then, work out x2(x – 2) and subtract from what you started with
2x2 – 4x
x – 2
x – 2 1D

21. You need to remember how to divide using Algebraic long division
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
2x x2 – 16x
2x3 – 8x2
3x2 – 16x
Second, divide 3x2 by x
= 3x
Then, work out 3x(x – 4) and subtract from what you have left
Third, divide -4x by x
= -4
Then, work out -4(x – 4) and subtract from what you have left
First, divide 2x3 by x
= 2x2
Then, work out 2x2(x – 4) and subtract from what you started with
3x2 – 12x
-4x
-4x -6 1D

22. Algebraic Fractions 19 ÷ 5 = 3 26 ÷ 3 = 8 1D
You need to remember how to divide using Algebraic Long Division
But, how do we deal with the remainder? 4
The ‘remainder’ is the numerator
19 ÷ 5
= 3 5
5 divides into 19 3 whole times…
The ‘divisor’ is the denominator
Another way to think of this sum is 
19 = (3 x 5) + 4 2
The ‘remainder’ is the numerator
26 ÷ 3
= 8 3
3 divides into 26 8 whole times…
The ‘divisor’ is the denominator
Another way to think of this sum is 
26 = (8 x 3) + 2 1D

23. Algebraic Fractions 2x2 + 3x - 4 x - 4 2x3 - 5x2 – 16x + 10 2x3 – 8x2
You need to remember how to divide using Algebraic Long Division
We did this division earlier 
So the sum we have including the remainder is:
2x2 + 3x - 4
x - 4
2x x2 – 16x
2x3 – 8x2
3x2 – 16x
3x2 – 12x
Remainder Divisor
-4x
-4x
- 6
2x x2 – 16x ÷
(x – 4) =
2x x - 4 +
x - 4 -6 6 =
2x x - 4 -
x - 4 1D

24. Algebraic Fractions x2 + 3x - 3 x - 1 x3 + 2x2 – 6x + 1 x3 – x2
You need to remember how to divide using Algebraic Long Division
Write (x3 + 2x2 – 6x + 1) ÷ (x – 1) in the form:
 Just do the division as normal… x2 + 3x - 3
x - 1
x x2 – 6x + 1
x3 – x2
3x2 – 6x + 1
3x2 – 3x
-3x + 1
-3x + 3 -2
Multiply both sides by (x – 1) 1D

25. 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