# Core 4 Notes Edexcel

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

HideShow resource information
• 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.
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

## Other pages in this set

Here's a taster:

Coor
dinat
e Geomet
ry
A
pa r
ame t
ri
c equation
of a
curve
is
one whi
ch does
not
give
the
rel
ati
onship
bet
w een
x and y
direct
ly
but
rather
uses
a t
hir
d var
iabl
e,
typi
cal
ly
t,
t
o do
so.
The
t
hir
d var
iabl
e
is
kno wn as
the
parame t

### Page 3

Here's a taster:

Bi
nomi
nal
Expansi
on
The
previ
ous ver
sion
of
the
binomial
t
heorem onl
y wor
ks
when
n i
s a
posi
ti
ve
i
nteger
.
If
n
is
any f
ract
ion,
t
he bi
nomial
theor
em becomes:
2 3
n(n-1)x
(1 + x)n = 1 + nx
1! + 2! + n(n-1)(n-2)x

Here's a taster:

I
mpl
ici
t
D i
ff
erent
iat
ion
N orma l
ly,
w he n
dif
ferent
iat
ing,
it
i
s deal
ing
w i
th
`expl
ici
t
funct
ions'
of
x,
where
a
valu e
of y
is
d efi
ned only
in
terms of

### Page 5

Here's a taster:

Di
ff
erent
iat
inga
Thi
s f
unct
ion
descr
ibes
growth
and decay,
and
it
s der
ivat
ive
giv
e s
a
measur
e of
t
he r
ate
of
change
of
thi
s gr
ow t
h/decay.
Si
ncey = ax
,
taki
ng
logs
of
bot
h si
des
gives
ln y = ln ax = xln a .…read more

Here's a taster:

The
key
to
d oi
ng t
he se
probl
ems i
s t
o i
denti
fy
thr
ee comp onent
s and
w r
it
e t
hem
down
ma t
hema ti
call
y:
Wh at
you
a r
e gi
ven
Wh at
i
s r
eq ui
red
Wh at
i
s t
he connecti
on
between t
he two
it
ems

### Page 7

Here's a taster:

C
Usi
ng
the
rever
se
of
the
chai
n r
ule,
t
he
fol
lowi
ng
gener
ali
sat
ions
can
be
found:
n
(ax+b)
(ax + b)n dx = 1
a n+1 +C
eax+b dx = 1
ae
ax+b + C
ax1+b dx = 1
a ln |ax + b| + C
cos (ax + b)dx = 1
a sin (ax + b) + C
sin (ax + b)dx =- 1
a cos (ax + b) + C
sec2(ax + b)dx = 1…read more

### Page 8

Here's a taster:

I
f
necessar
y,
use
u=f(
x)
to
change
the
val
ues f
or
the
li
mit
s of
i
ntegr
ati
on
Put
your
x'
s
back
in
agai
n at
t
he end
and
fi
nish
up.
Exampl
e
3
Suppose
we
wis
h
to
fi
nd
(9 + x)2dx
1
We
make
the
subst
it
uti
on
u = 9 + x .…read more

Here's a taster:

Car
e must
be
taken
over
t
he
choi
ce
of
uand
dx .
The
aim
is
to
ensur
e t
hat
i
t
is
si
mpl
er
to
int
egr
ate
thanv du
dx .
So
choose
u t
o be
easy
to
dif
fer
ent
iat
e and
dv
dx t
o
be
easy
to
int
egr
ate.
General
ly,
t
his
me ans
choose u
to
be t
he simp l
er
of
the
two f
u nc ti

### Page 10

Here's a taster:

Di
ff
erent
ial
Equat
ions
I
n gener
al
a di
ff
erent
ial
equat
io n
ma y ha ve x an d
y t
erms
on bot
h si
des,
but
i
f
the
dy
equat
ion
is
of
a
cert
ain
form
f (x)g(y) dx = h(y)i(x)
,
w e
can
rear
range
to
have al
l
t
erms
incl
uding
x on t
he
ri
g ht
ha nd
side and
all
t
erms
incl
udi
ng y