introduction - loreto college, st albans · 3( 2) 2( 2) x x algebraic fractions you need to be able...

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.

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)


Factor (4)

3

2 6

x

x

3

2( 3)

x

x

Factorise the Denominator

Factorise the Denominator


Factor (x + 3)


Factor (x + 3) 1

21A

3( 2)

2( 2)

x

x

Algebraic Fractions




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


Denominator by 6

Factorise

Divide by (x + 2)

3

2

Divide by (x + 2)

1A

Algebraic Fractions




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

Algebraic Fractions




Another Example of a Fraction within a Fraction…

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

Example Questions

1

1

xx

x


Denominator by x

1A


Denominator by x

Factorise Factorise

2

2

1x

x x

1x

x

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

1x

x x

11

x

Split the Fraction up

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

1B

1 3

2 5

3

10

a c

b d

ac

bd

a)

b)

3c)

5

5

9

15

45

1

3

3

5

5

9

1

3

1

31

Algebraic Fractions







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 Denominator

Algebraic Fractions







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

Algebraic Fractions









second Fraction.

Example Questions

1B

a a

b c

c

b

b)

a c

b a

1

1

Algebraic Fractions









second 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 Fractions

You 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 Numerators

Add the Numerators

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 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 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

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 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.

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 4 5

= 8 2 3

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

introduction - loreto college, st albans · 3( 2) 2( 2) x x algebraic fractions you need to be able...

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