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