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.