Algebraic Fractions

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 FractionsYou 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

16

20

Example Questions

4

5

Divide by the common Factor (4)

Divide by the common Factor (4)

3

2 6

x

x

3

2( 3)

x

x

Factorise the Denominator

Factorise the Denominator

Divide by the common

Factor (x + 3)

Divide by the common

Factor (x + 3) 1

21A

3( 2)

2( 2)

x

x




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

12

1 23 3

1x

x

3 6

2 4

x

x

Multiply the Numerator and Denominator by

6

Factorise


6

Factorise

Divide by (x + 2)

3

2

Divide by (x + 2)

1A




every term.

Sometimes you will have to Factorise both the Numerator and

Denominator.

Example Questions

2

2

1

4 3

x

x x

( 1)( 1)

( 1)( 3)

x x

x x

Factorise the Numerator AND

Denominator

1A

Factorise the Numerator AND

Denominator

1

3

x

x

Divide by (x + 1) Divide by (x + 1)

( 1)( 1)

( 1)

x x

x x




every term.

Another Example of a Fraction within a Fraction…

You will usually be told what ‘form’ to leave your answer in…

Example Questions

1

1xx

x

Multiply the

Numerator and Denominator by

x

1A


x

Factorise Factorise

2

2

1x

x x

1x

x

Divide by (x +

1)Divide by (x +

1)

1x

x x

11x

Split the Fraction up


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

1B

1 3

2 5

3

10

a c

b d

ac

bd

a)

b)

3c)5

5

915

45

1

3

3

5

5

9

1

3

1

31


multiply and divide Algebraic Fractions


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

1B

a c

b a

c

bd)

1

1

e) 2

1 3

2 1

x

x

1 3

2 ( 1)( 1)

x

x x

3

2( 1)x

1

1

Factorise

Multiply Numerator

and Denominato

r

Algebraic FractionsYou need to be able to multiply and divide Algebraic Fractions


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

1B

5 1

6 3

15

6

a)

5 3

6 1

5

2









Fraction.

Example Questions

1B

a a

b c

c

b

b)

a c

b a

1

1









Fraction.

Example Questions

1B

2

2 3 6

4 16

x x

x x

c)

22 16

4 3 6

x x

x x

2 ( 4)( 4)

4 3( 2)

x x x

x x

( 4)

3

x

Leave, Change and Flip

Factorise

Multiply the Numerators

and Denominators

1

1

1

1

Algebraic FractionsYou need to be able to add

and subtract Algebraic Fractions


When adding and subtracting fractions, they must first have the same Denominator. After that, you just add/subtract the Numerators.

Example Questions

1C

a)1 3

3 4

4 9

12 12

13

12

Multiply all by 4

Multiply all by 3

Add the Numerator

s

Add the Numerator

s

Algebraic FractionsYou need to be able to add

and subtract Algebraic Fractions



Example Questions

1C

ab

x

Imagine ‘b’ as a

Fraction

Example Questions

b)

1

a b

x

a bx

x x

a bx

x

Multiply all by x

Combine as a single Fraction

Combine as a single Fraction

Algebraic FractionsYou need to be able to add and

subtract Algebraic Fractions



Example Questions

1C

2

3 4

1 1

x

x x

Factorise so you can compare

Denominators

c)

3 4

1 ( 1)( 1)

x

x x x

( 1)

( 1

3 4

( 1) ( 1)( 1) )

x

xx x

x

x

3 3 4

( 1)( 1)

x x

x x

7 3

( 1)( 1)

x

x x

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

Third, Divide -2x by x

-2

We then subtract ‘-2(x – 3) from what we have left

Algebraic Fractions

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

1D

x - 3 x3 + 2x2 – 17x + 6

x2

x3 – 3x2

5x2 - 17x + 6

5x+

5x2 - 15x

- 2x + 6

2-

- 2x + 6

0

First, Divide x3 by x

x2

We then subtract ‘x2(x – 3) from what we started with

Second, Divide 5x2 by x

5x

We then subtract ‘5x(x – 3) from what we have left

Algebraic FractionsYou 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

1D

x - 2 x3 + 0x2 – 3x - 2

x2

x3 – 2x2

2x2 – 3x - 2

2x+

2x2 – 4x

x – 2

1+

x – 2

0

First, divide x3 by x

= x2

Then, work out x2(x – 2) and subtract from what you started with

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

Algebraic FractionsYou 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.

1D

x - 4 2x3 - 5x2 – 16x + 10

2x2

2x3 – 8x2

3x2 – 16x + 10

3x+

3x2 – 12x

-4x + 10

4-

-4x + 16

-6

First, divide 2x3 by x

= 2x2

Then, work out 2x2(x – 4) and subtract from what you started with

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

Algebraic Fractions

1D

You need to remember how to divide using Algebraic

Long Division

But, how do we deal with the remainder?

19 ÷ 5

26 ÷ 3

= 3

45

= 8

23

5 divides into 19 3 whole

times…

The ‘divisor’ is the

denominator

The ‘remainder’ is the

numerator

3 divides into 26 8 whole

times…

The ‘divisor’ is the

denominator

The ‘remainder’ is the

numerator

Another way to think of this sum is 19 = (3 x 5) +

4

Another way to think of this sum is 26 = (8 x 3) +

2

Algebraic Fractions

1D

x - 4 2x3 - 5x2 – 16x + 10

2x2

2x3 – 8x2

3x2 – 16x + 10

3x+

3x2 – 12x

-4x + 10

4-

-4x + 16

-6

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:

2x3 - 5x2 – 16x + 10

÷ (x – 4) 2x2 + 3x - 4= + - 6

x - 4

2x2 + 3x - 4= - 6

x - 4

Remainder Divisor

Algebraic Fractions

1D

x - 1 x3 + 2x2 – 6x + 1

x2

x3 – x2

3x2 – 6x + 1

3x+

3x2 – 3x

-3x + 1

3-

-3x + 3

-2

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…

2 1Ax Bx C x D

3 22 6 1x x x 1x 2 3 3x x 2

1x

Multiply both

sides by (x – 1)

3 22 6 1x x x 2 3 3 1x x x 2

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

Date post:	22-Jan-2016
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Algebraic Fractions

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