www.mathcentre.ac.uk/.../Partial_Fractions.mp4

An algebraic fraction is one
where the numerator and

denominator, both polynomial
expressions.

This is an expression where
every term is a multiple of a

power of X, like.

5X to the 4th
plus 6X cubed plus

7X plus 4.

The degree of a polynomial is
the power of the highest

Terminix. So this is a
polynomial of degree 4.

The number in front of X in each
case is the coefficient of that

term. So the coefficient of X to
the 4th is 5. The coefficient of

X cubed is 6.

Now look at these fractions
X over X squared +2.

Or X cubed plus three
over X to the 4th plus

X squared plus one.

In both cases.

The numerator is a polynomial of

lower degree. Then the

denominator. X's against X
squared X cubed as against X to

the 4th week. All these proper

fractions. With other fractions,
the polynomial may be of higher

degree in the numerator. For
instance, X fourth plus X

squared plus X.

Over X cubed plus X +2 or
it may be of the same degree.

Such as X plus four over
X plus three. We call these

improper fractions. Down,
look like to look at

how we add and subtract
fractions. Take for instance

these two fractions.

In order to add these two
fractions together, we need to

find the lowest common
denominator. In this particular

case it's X minus 3 *
2 X plus one, so we

say that this sum is the

equivalent of. In the
denominators we are going to

have X minus 3 * 2 X plus one.

In order to get from there to
there, we multiplied by 2X plus

one, we've multiplied the
denominator by 2X plus one. So

we must multiply the numerator
by 2X plus one.

And in order to get from here to
here, we've multiplied the

denominator by X minus three. So
we've got to multiply the

numerator by X minus three, and
this gives us just X minus

three. Now we need to collect

up. The denominators of the
same, so we can just right.

X minus 3 * 2 X Plus
One and on top we have 2

* 2 X is 4X Minus, X
gives us 3X.

And we also have 2 * 1 is 2.

Minus minus three is plus three,
so 2 + 3 is +5 and that

is the answer to that some.

Sometimes in mathematics we need
to do this operation in reverse.

In calculus, for instance, or
when dealing with the binomial

theorem. We sometimes need to
split a fraction up into its

component parts, which are
called partial fractions. Let's

take the sum that I've just
dealt with. We got the answer.

Three X +5.

Over.

X minus 3 * 2
X plus one.

So how do we get this back to
its component parts? Well?

We only have two factors in the
denominator, X minus three and

2X plus one.

So. It must be something
over X minus three plus

something. Over 2X plus one.

And what are these some things?
They can only be plain numbers,

because if they involved X or
powers of X then these would be

improper fractions, so we're
quite entitled to say that 3X

plus five over X minus 3 * 2 X
Plus one is a over X minus three

plus B over 2X plus one where
A&B are just plain numbers.

The next thing to do is to
multiply everything through by

what's on the bottom X minus 3 *
2 X plus one.

If you multiply the left hand
side by that, we just get three

X +5 equals.

A over X minus three times X
minus 3 * 2 X plus one the

X minus threes will cancel,
and we're just left with a

Times 2X plus one.

B over 2X Plus One Times X minus
3 * 2 X Plus one. This time the

2X plus ones will cancel and we
just left with B Times X minus

three. Now this is an identity,
which means that it is true for

all values of X.

If this is so, then we can
substitute special values for X

and it will still be true.

For instance, if we make X equal
to minus 1/2.

This bracket will become zero
and a will disappear.

If we make X equal to three,
this bracket will become zero

and be will disappear.

And I'm going to do just that.

If X equals minus 1/2.

We get three times minus 1/2
is minus three over 2 +

5. That is 0.

Equals B times minus
1/2 - 3.

This is just Seven over
2 and we get 7 over 2

equals. This is minus 7
over 2 - 7 over 2B so

B is equal to minus one.

All right, this line
in again 3X plus

5. Equals a Times
2X plus one.

Plus B times.

X minus three.

This time I want to try and find
a, so I'm going to put X equal

to 3. If X equals

3. We have 3 threes and
9 + 5 is 14.

3266 plus One is 7.

That is going to be 0, so be
will disappear, so A is equal to

14 / 7. In other words, a IS2.

We already had the
equal to minus one.

So what do we have now?

I had three X +5 over.

X minus three.

2X plus one times 2X plus
one equals a over.

X minus three plus B over 2X
plus one. Since A is 2 and

B is minus one, we can see
that this is 2 over X minus

three plus, sorry minus.

One over 2X Plus One, which is
the sum that we started with

and we have now broken this
back into its component parts

called partial fractions.

Do another example. Let's say
that we have to express

3X over X minus one
times X +2 in partial

fractions. Again, we look at the
denominator. The factors in the

denominator X minus one and X
+2. So we say that this

expression is equal to a over X
minus one plus B over X +2.

We multiply through by X minus
one times X +2 on the left hand

side. This just gives us 3X.

On the right hand side, a over X
minus one times X minus one

times X +2 X minus ones cancel
out, and we're left with a Times

X +2. Be over X +2
times X minus 1X Plus 2X

plus Two's cancel out and
we're left with B Times X

minus one.

This time the special values
that I'm going to take our X

equals minus two because that
will make that zero and thus

eliminate A and X equals 1,
which will make that zero and

thus eliminate B.

If X equals minus

2. We get three times
minus two is minus 6.

That is 0, so a disappears.

Minus 2 - 1 is minus three,
so this is minus 3B.

So.

B equals minus 6 divided
by minus 3 equals 2.

Alright, this
expression in

again.

This time I'm
going to put

X equal to

1. 3 * 1

is 3. 1 + 2
is 3, so we get 3A.

1 - 1 is 0 so be disappears.

If 3A equals 3, then a is
going to equal 1, so we've

got a equal 1. We already
had B equal to two.

I'm not going to write the whole

expression in again. We have 3X.

Over X minus one
times X +2 equals.

One over X minus one because a
is 1 + 2 over X +2 because

be is 2 and that is the answer.

Sometimes the denominators more
awkward, for example, to

express 3X plus one
over X minus one

squared times X +2.

There are actually three
possibilities for a denominator

in the partial fraction.

We've got X minus One X +2, but
there's also the possibility of

X minus 1 squared.

So we write down a over
X minus one plus B over

X minus 1 squared.

Plus C over X
+2.

Again, we multiply through by
the bottom line here, so we get

a over X minus one times X
minus one squared times X +2.

One of the X minus ones will
cancel, leaving us with 3X plus

one equals a Times X minus one
times X +2.

B over X minus one squared times
X minus one squared times X +2.

Both of the X minus one squared
will cancel, leaving us with B

Times X +2.

And then we have C over X +2
times X minus one squared times

X +2. This time the X +2 is will
cancel, leaving us with C Times

X minus 1 squared.

Again, the special values X
equals one will make this zero,

so a will disappear and it will
make this zero. So see will

disappear. If X equals one, we
have 3X Plus One is 4.

That zero so that expression

disappears. 1 + 2 is 3, so
we have 3B.

This is 0, so this disappears.
So we have 4 equals 3B. Giving B

equals 4 over 3.

If X equals.

Minus 2.

We have minus 2 * 3 is minus 6
Plus One is minus 5.

Equals this is 0, so this

disappears. This is 0, so this
disappears minus 2.

Minus one is minus 3 squared is
9, so we have minus five is

9C, which gives us C is minus

5. Over 9.

We now need to find a.
I'm just going to write this

expression out again.

I've
written

the.
Expression following, see out

like that because in a minute
I'm going to multiply it out.

Unfortunately, there's no
special value of X that will

eliminate B&C. To give us A.

We can use any special value. We
could use X equals 0. This would

give us an equation in AB&C
since we already know be in. See

this would give us a.

But I'm going to use a
different technique, one

called equating
coefficients, and to do

that I've got to multiply
this lot right out.

So we get equals a.

And we have an X Times X for X

squared. We have a minus 1X plus
2X, so that gives us Plus X.

And we have minus 1 * 2 which
gives us minus 2.

And then plus BX

+2. Plus C.

X times X is X squared. We have
a minus X under minus six, so

that's minus 2X and then minus
one times minus one is plus one.

I'm not going to collect up all
the terms. For instance, we have

an A Times X squared here and we
have a C Times X squared here.

So we have a plus C Times X

squared altogether. We also have
an A Times XAB Times X&A minus

two C Times X.

A+B minus two
C Times X?

And finally we have minus 2A.

2B and C.

So the constant becomes minus
2A plus 2B Plus C.

Now. So we have 3X plus one

equals. This line.

But in this line, we have a

Turman X squared. 3X
plus one doesn't have

anything in X squared.

But this is an identity. It must
be true for all values of X, and

the only way that this can be
true is for A plus E to be 0 so

that X squared disappears on
this side. So we can say that a

plus C equals 0.

We already know that C is minus
5 over 9, so in order for 8 plus

C to be 0.

A must be plus five over 9.

And we already worked out B as
being equal to.

For over 3, this means that we
can write out the solution to

the whole problem.

3X plus one over
X minus one squared

times X +2.

Equals. A5 over 9X
minus one plus B is

4 over 3 four over
3X minus 1 squared.

See is minus 5 over 9, so
we have minus five over 9 X

+2. Another case
we must consider.

Is where the denominator
contains a quadratic that can't

be factorized as in 5X over.

X squared plus X Plus One
Times X minus 2.

If we to express this in partial
fractions, the two denominators

are going to be X squared plus X
Plus One and X minus 2.

When the denominator is X
squared plus 6 plus one, we have

to consider the possibility that
the numerator can contain a

termine ex, because the
numerator would still be of

lower degree than the
denominator, and this would

still therefore be a proper
fraction. So we write a X plus B

over X squared plus X plus one.

Plus C over X minus two
as before. We multiply this out

so we get that five X
equals X plus B Times X

minus 2. Plus
C Times

X squared.
Plus 6 + 1.

One special value we can use is

X equals 2. And if.

X equals 2, we

get 5X5210. This is 0,
so this all disappears and we

get 2 twos of 4 + 2 is 6 plus
one is 7, so 10 equals 7 C.

Giving C equals 10 over 7.

Unfortunately, there's no value
for X would enable us to get rid

of C, so we're going to have to
use the technique of equating

coefficients. I'll write
this out again.

In order
to equate

coefficients, I'm
going to

have to
multiply this

out.

X times X is X squared.

X times minus two is minus two

AX. B times X
is BXB times minus two

gives us minus 2B Plus
CX squared Plus CX Plus

C. Again, I'm going to collect
like terms. So for instance for

X squared we have.

AX squared and CX squared.
So we have a plus

CX squared for X. We
have a minus two AAB&C.

So minus two A+B Plus
CX and for a constant

we have minus 2B Plus

C. We still
need to find

both A&B.

For two unknowns we need 2
equations, so we are going to

have to solve for two different

coefficients. Now the left hand
side is just 5X, so there is no

coefficient in X squared.

In order to eliminate X squared,
we can say that a plus C equals

0. We already know what see is
10 over 7. In order for a plus C

to be 0, this will make a minus
10 over 7.

The left hand side also has
no constant coefficient, so

that means that this
expression must be 0. So we

say minus 2B Plus C equals 0.

Giving us. C equals

2B. Or B equals C over
two, which gives us B as being.

5 over 7.

So we have a equal to
minus 10 over 7B equal to

five over 7 and C equal
to 10 over 7.

This means that 5X over.

X squared plus X plus one.

Times X minus two is equal to
a X which is minus 10 over

7X. Plus B, which is 5
over 7 all over X squared plus

X plus one.

Plus C, which is 10 over 7
over X minus two and are now

tidy. This up the Seven comes
down to be multiplied by the X

squared plus X plus one. So we
get minus 10X plus five over 7

X squared plus X Plus One plus
and again the Seven comes down

10 over 7X minus 2.

Equals and to finish it off we
need to take five out of this

expression as a factor, which
gives us five times minus 2X

plus one over 7 X squared plus X
plus 1 + 10 over 7X minus 2.

So far I've only dealt with
proper fractions where the

numerator is of lower degree
than the denominator. Now, like

to look at an improper fraction.

Let's Express.

4X cubed plus 10X
plus four over X

into 2X plus one.

In partial fractions.

The numerator is of degree 3.

The denominator, if you multiply
the X by the two X, you get 2 X

squared, so the denominator is

of degree 2. This means that
this is an improper fraction.

What this means is that if you
divide the numerator by the

denominator, you're going to be
dividing otermin X cubed by a

term in X squared.

So you could get a Terminix.

Which means that we have to
write down acts. We may also get

a constant term, so we have to
write down B.

Then we can do our fractions.

If I now multiply but
through I get a X

Times X Times 2X plus
one, so we get 4X

cubed plus 10X plus four

equals a. X squared

Times 2X plus one.

Plus BX times 2X

plus one. Plus C
Times 2X plus one.

Plus DX

Using special values.

If I use X equals 0, then
the term the D, the B, and

the A are all going to
disappear and I'm just left

with see. So if X equals 0.

X cubed is zero, X is zero. I
just get 4 equal to.

2X is 0, so it's just C, so we
have C equal to four. The other

special value is X equal to

minus 1/2. If X equals minus
Alpha, this is 0, so this will

disappear. This is 0, so this

will disappear. And this will
disappear, just leaving me with

D. So I get.

Minus 1/2. Cubed is
minus an eighth, so we

get minus four over 8.

Plus 10 times minus 1/2 inches
minus 10 over 2 + 4

equals. D times minus

1/2. I'll just
write that down again,

minus four over 8
- 10 over 2.

+4. Equals minus

1/2 D. Minus 4 over 8
is just minus 1/2.

Minus 10 over 2 is minus 5
+ 4 equals minus half D.

Minus 5 + 4 is minus one,
so I've got minus 1 1/2 equals

minus 1/2 D.

Minus 1 1/2 is just three
times minus 1/2, so this

gives us D equal 3.

Special values won't give me a
or be, so I'm going to have to

equate coefficients. This means
I have to write this expression

out again. 4X
cubed plus

10X. +4 equals
a X squared times.

2X plus one plus
BX times 2X plus

one. Plus

C times 2X plus one
plus DX.

I'm now going to multiply
this out.

X squared times
2X is 2A X cubed.

X squared times one
is just X squared.

This gives me 2B X squared.

This gives me

BX. This gives Me 2

CX. This gives
me C.

And then.

Plus DX And collecting terms, we
only have one Turman X cubed, so

that is just 2A X cubed.

Plus we have two terms in X
squared, A and 2B.

We have three terms
in XB2C and D.

And finally,
the constant

term see.

Now look at the Turman X cubed.
We have 4X cubed on the left.

And two AX cubed on the right.
This means that 2A must be equal

to 4. Giving us a equal to two
now look at the Turman X

squared. There is no Turman X
squared on the left.

And on the right
we have a plus 2B.

This means that as there isn't
Turman X squared on the left, a

plus 2B must be equal to 0.

So we have a plus 2B

equals 0. Which means that.

A equals minus

2B. Which means
that B equals.

Minus two over 2
equals minus one.

I'll just write those
values in again.

A equals 2.

B equals minus one C
equals 4D equals 3.

So if we take our original
expression 4X cubed plus 10X

plus four over X times.

2X plus one. This is equal
to axe, so 2X.

Minus B. Plus see over X,
so that's four over X Plus D

over 2X Plus One which is 3 over
2X plus one.