# Core 4 Notes Edexcel

Notes for Core 4 maths, done for the edexcel syllabus but should be helpful with other exam boards.

- Created by: tom
- Created on: 07-06-11 19:40

First 1033 words of the document:

Par

ti

al

Fract

ions

(x-3)(2x-1)

3x+5

An

algebr

aic

fr

act

ion

such

as

can

be

broken

down

int

o si

mpl

er

par

ts

A B

cal

led

par

ti

al

fr

act

i

o ns

so

that

i

t

is

in

the

for

m (x-3) + (2x-1)

Once

a fr

acti

on has

been

spl

it

i

nto

it

s const

it

uent

s,

it

can

be

used

in

int

egr

ati

on

and

binominal

t

heorem.

Spl

it

ti

ng

Int

o Par

ti

al

Fract

ions

Part

ia l

Fracti

ons

can

be

spl

it

up

in

two

w ays:

substitution

or

equating

coefficients.

Subst

it

uti

on :

Thi

s i

s used

for

gener

al

algebr

aic

fr

act

ions

w i

th

two

or

thr

ee

fact

ors,

wi

thout

a

r

epeated denominat

or.

1.Th e fi

rst st

e p

in

th i

s p r

o c ess

w oul

d be

to

ma ke

the

algebrai

c f

ract

ion

equal

partial

fraction s w it

h

a l

l p ossi

ble

denominator

s,

A and

B

as const

a nt

nume rators .

2.Mu lti

p l

y

the

n ume rato r

o f

A ,

by

the

de nomi

nator

of

B.

T hen

ma ke t

hese

equal

t

o th e nu me ra t

o r of

th e o r

igi

nal

expressi

on.

It

wi

ll

now be

in

the f

orm of

6x - 2 = A(x + 1) + B(x - 3)

3.Substi

tut

e i

n val

ues

of

x t

hat

w i

l

l

ma ke

one

of

the

brack

e t

s z

e r

o.

Then

use

t

his

to

w or

k out

t

he

value

of

A

and B.

Then

repl

ace

them as

the numer

ator

s

and

you have

your

part

ial

f

ract

ions.

Equati

ng Coeff

ici

ents:

Thi

s f

oll

ows t

he

same

fi

rst

t

w o

steps

as subst

it

uti

on,

however

someti

me s,

subst

it

uti

on wi

ll

not

work.

I

n t

h i

s case,

you

can equat

e t

he

coef

fi

cient

s of

x

and

t

he const

ants

to work

out

A and B.

1.Follow th e

first

two st

ep s

of

subst

it

uti

on

2.Exp an d

th e brac ket

s so

that

you

end

u p

wit

h somet

hing

in

the

for

m

Ax + A + Bx - 3B

3.Make t

he coef

fi

cient

of

x

equa l

t

o (

A +B)x.

Thi

s gi

ves

yo u

one si

mp l

e

equat

ion.

4.Do t

he same t

hing

for

t

he

constant

s (A 3B)

and

the const

ant

o n

the

end

of

t

he or

igi

nal

expressi

on.

You

now

have t

w o

simple

equati

o ns

that

can

be

sol

ved

u si

ng

simult

aneous equati

ons.

Bot

h of

t

hese

techni

ques

can

be used

w hen

the

fr

act

ion

has

mo r

e t

han

two

f

act

ors

(i

e.

use

A ,

B

and

C )

or

one

that

has

a r

epeated

li

near

f

act

or.

An

algebr

aic

fr

act

ion

is

improper

w hen

the

degr

ee

of

the

numer

ator

i

s

equal

t

o,

or

lar

ger

t

han,

t

he

degr

ee

of

the

denomi

nat

or.

An

improper fraction

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