In simplifying algebraic fractions, we occasionally need
a process known as.
Polynomial. Division.
Before we do that, I want to take you back to something
that you actually know very well indeed, and that's
ordinary long division.
You know how to do long division, but I want to go over
it again. 'cause I want to point out certain things to you.
'cause the things that are important about long division
are also important in polynomial
division so. Let's have a look at a long division. Some
supposing I want to divide 25.
Into Let's say
2675.
When I would have to do is look at 25 in tool 2.
No way 25 into 26. It goes once and write the one there.
Add multiply the one by the 25.
And subtract and have one left.
Hope you remember doing that. You were probably taught how to
do that at primary school or the beginnings of Secondary School.
Next step is to bring down the next number, so we bring down
17. Well, we bring down Seven to make it 17 and now we say how
many times does 25 going to 17.
It doesn't go at all. It's not enough, so we have to record the
fact that it doesn't go with a
0. Next we bring down the five. So now we've got 175 and
we say how many times does 25 go into that? And it goes 7
and we can check that Seven 535, five down three to carry. 7 twos
are 14 and three is 17.
Tracked, we get nothing left.
So this is our answer. We've nothing left there, no
remainder, nothing left over. And there's our answer.
2675 divides by 25 and the answer is 107. They just look at
what we did. We did 25 into 26 because that went.
We then recorded that once that it went there, multiplied, wrote
the answer and subtracted.
We brought down the next number.
Asked how many times 25 went into it, it didn't go. We
recorded that and brought down the next number. Then we said
how many times does 25 going to that Seven we did the
multiplication, wrote it down, subtracted, got nothing left
so it finished.
What we're going to do now is take that self same process and
do it with algebra.
So let us
take. This 27 X cubed.
+9 X squared.
Minus 3X. Minus 10.
All over.
3X minus 2.
We want to divide that into that. We want to know how many
times that will fit into there, so we set it up exactly like a
long division. Problem by dividing by this. This is what
we're dividing into 27 X cubed plus nine X squared minus three
X minus 10.
So we ask ourselves, how many times does well? How many times
does that go into that? But difficult what we ask ourselves
is how many times does the excpet go into this bit?
Just like we asked ourselves how many times the 25 went into the
26, how many times does 3X?
Go into 27 X cubed. The answer must be 9 X squared because
Nynex squared times by three X gives us 27 X cubed and we need
to record that. But we need to record it in the right place and
because these are the X squared's we record that above
the X squares.
So now we do the multiplication.
Nine X squared times 3X is 27
X cubed. Nine X squared times minus two is minus 18 X squared.
Just like we did for long division, we now do the
Subtraction. 27 X cubed takeaway 27 X cubed none of
them, because we arrange for it to be so Nynex squared
takeaway minus 18 X squared gives us plus 27 X squared.
Now we do what we did before we bring down the next one, so we
bring down the minus 3X.
How many times does 3X go into 27 X squared?
Answer. It goes 9X times and we write that in the X Column.
So now we have 9X times 3 X 27 X squared 9X times, Y
minus 2 - 18 X.
And we subtract again.
27 X squared takeaway, 27 X squared, no X squared, but we
arrange for it to be like that, minus three X minus minus 18X.
Well, that's going to give us plus 15X altogether, and we
bring down the minus 10.
3X into 15X. This time it goes five times, so we can say plus
five there. And again it's in the numbers. The constants
column at the end.
Five times by 15 times by three X gives us 15X. Write it down
there five times by minus two gives us minus 10 and we can see
that when we take these two away. Got exactly the same
expression. 15X minus 10 takeaway. 50X minus 10 nothing
left. So there's our answer, just as in the long division.
The answer was there.
It's there now so we can say that this expression is equal to
9 X squared plus 9X.
Plus 5. Let's take another one.
So we'll take X to the 4th.
Plus X cubed.
Plus Seven X squared minus six X +8.
Divided by all over X squared, +2 X
+8. So this is what we're dividing by and this is what
we're dividing into is not immediately obvious what the
answer is going to be. Let's have a look X squared plus 2X
plus 8IN tool.
All of this.
Our first question is how many times does X squared going to X
to the 4th? We don't need to worry about the rest, we just do
it on the first 2 bits in each one, just as the same as we did
with the previous example. How many times X squared going to X
to the four will it goes X squared times? So we write it
there over the X squared's. Now we do the multiplication X
squared times. My X squared is X to the 4th.
X squared by two X is plus 2X cubed X squared by 8 is
plus 8X squared.
And now we do the Subtraction X. The four takeaway X to the 4th
there Arnold, but we arranged it that way. X cubed takeaway 2X
cubed minus X cubed. Seven X squared takeaway, 8X squared
minus X squared and bring down the next term.
Now we say how many times does X squared going to minus X cubed,
and it must be minus X, and so we write it in the X Column.
And above the line there, next the multiplication minus X times
by X squared is minus X cubed minus X times 2X is minus two X
squared and minus X times by 8 is minus 8X.
Do the subtraction minus X cubed takeaway minus X cubed. No ex
cubes minus X squared minus minus two X squared or the minus
minus A plus, so that effectively that's minus X
squared +2 X squared just gives us X squared.
Minus six X minus minus 8X. Well, that's minus 6X Plus 8X
gives us plus 2X and bring down the next one.
X squared plus 2X plus a 12 X squared goes into X squared
once. And so X squared plus 2X plus eight. And again we
can see these two are the same when I take them away,
I will have nothing left and so this is my answer.
The result of doing that division is that.
Well, the one that started us off on doing this was if
you remember.
X cubed minus one over X minus one.
This looks a little bit different, doesn't it? Because
whereas the space between the X Cube term and the constant term
was filled with all the terms?
This one isn't.
How do we cope with the?
Let's have a look. Remember, we know what the answer to this one
is already. So what we must do is right in X cubed and then
leave space for the X squared term, the X term and then the
constant term. So what I asked myself is how many times does X
go into XQ, and the answer goes in X squared. So I write the
answer there where the X squared term would be.
X squared times by X is X cubed.
X squared times by minus one is minus X squared.
And subtract X cubed takeaway X cubed no ex cubes.
0 minus minus X squared is plus X squared.
Bring down the next term. There is no next term to bring down.
There's no X to bring down.
So it's as though I got zero X. There was no point in writing
it. If it's not there, so let's carry on X in two X squared that
goes X times. So record the X there above where the X is would
be. Let's do the multiplication X times by X. Is X squared.
X times Y minus one is minus X. Do the subtraction X squared
takeaway X squared is nothing.
Nothing takeaway minus X. It's minus minus X.
That gives us Plus X.
Bring down the next term. We have got a term here to bring
down it's minus one.
How many times does X going to X? It goes once.
Long times by XX. One times by minus one is minus one. Take
them away and we've got nothing left there and so this is my
answer X squared plus X plus one, and that's exactly the
answer that we had before. So where you've got terms missing?
You can still do the same division. You can still do the
same process, but you just leave the gaps where the terms would
be and you'll need the gaps because you're going to have to
write something. Up here in what's going to be the answer.