When we come across fractions, one of the things that we have
to do is to look at them and see if we can put them in a simple
form. In fact, see if we can put them in a form that might be
called the lowest terms. So for instance, if we have a fraction
12 over 36, then what we want is a fraction in its lowest terms.
Now 12 will divide into both 12 and 36, so we can divide 12 into
12 one and 12 into 36 three. So that reduces to the fraction 1/3
the two fractions are the same.
They are equivalent fractions. Now that's OK with numbers, but
we want to be able to do it with similar things, but with algebra
in other similar expressions that have got letters in them.
And where is here? We look for a number that was in fact a common
factor. That's a number that will divide into both the top
and the bottom. What we've now got to look for is a common
factor. That's an expression that will divide into both the
top and the bottom. So instead of talking about it, let's have
a look at some examples. So supposing we've got 3X cubed all
over X to the 5th.
We've got to look at is what's the same on the top and on the
bottom? Well, we've got a 3 here, but no numbers down here.
We've got X is here and we've got X is here and what we can
see is that we've got X cubed here and X to the fifth here.
Now X cubed times by X squared gives us X to the 5th, so any
fact we've got a common factor of X cubed on the top and on the
bottom. So let's just write that down so we can see it more
clearly. Thanks for the 5th is X cubed times by X squared? What
we can see very clearly here is we've got a common factor of X
cubed, so we can divide top and bottom by X cubed. That leaves
us with three over X squared. Now we don't need to do this
middle step every time we have to be able to do is to see
exactly what that common factor
is. So let's have a look at some examples and the examples will
increase in terms of their complexity as we move through
them in terms of the difficulty. So what's common here to both
the top and the bottom? On the top? We can see we've got X
cubed, and on the bottom X squared, and we know that if we
multiply X squared by X, it will give us X cubed. So in effect we
can divide. Top and bottom by X squared X squared in two X
squared goals once and X squared into X cubed leaves us with X
because X Times X squared gives us X cubed. Similarly, Y into Y
goes once and Y into Y cubed. Well why times by Y squared
gives us Y cubed. So if we divide in what we get there is Y
squared. So on the top we've got X times by.
Squared on the bottom. We've got one times by one so we don't
really need to write the one there. And there's our answer.
You can also involve numbers in this, so we have alot minus
16 X squared Y squared over 4X cubed Y squared.
Here minus 16 an 4 four goes into minus 16 - 4
times. X squared on the top and X cubed underneath X squared
goes into X squared. Once an X squared goes into X cubed X
times and here Y squared is the same on both top and bottom, so
Y squared into Y squared goes once and Y squared into Y
squared ones as well.
So here we minus 4 * 1 * 1 - 4 over.
14 went into 4 one select remember one times X times
one that's X. So we have minus four over X.
Sometimes that common factor may not be obvious. We may have to
work to see it. We may have to work so that it stands out over
what we've got. So let's have a look at something like this.
X squared minus two XY all
over X. Well, is there a
common factor? The best way to look at this is not so much. Is
there a common factor on top and bottom, but is there a common
factor here? Can we factorize
this expression? Well, in each term that is at least an X as an
X here and an X squared here. So we can take that X out as a
common factor X times by X gives us the X squared.
Taking X out of here, we're left with minus two Y so that X times
Y minus two Y would give us minus two XY, and then that's
all over X. Now we can cancel the X. We can divide the top a
numerator by X and the bottom by X&X in 2X goes one and X in 2X
goes once, and So what we're left with is one times that
divided by one. So we just left.
With X minus two Y.
Have a look at another One X to the power 6.
Minus Seven X to the fifth plus 4X to the
8th. All over X squared. Again, what we want to do is look at
this top line and see. Have we got a common factor.
Well, yes, we have its X to the power five 'cause X the
power five is included in X to the power 6 and X to the power
8. So we can take that out as a common factor X to the power
five times by X will give us X to the power 6.
Minus Seven X to the power five. We've got the X to the power 5
outside, so we want minus 7.
Plus X to the power 8.
We're taking out X the power 5th, so it's going to be 4X to
the power three. Close the bracket and all over X squared.
Now we can see what we're looking at is. Is there
something common now between these two? And clearly we can
divide X squared by X squared and we can divide X to the power
5 by X squared.
So we'll do that X in two X
squared once. X squared into X to the power five is X cubed.
And so we end up with X cubed times by X minus 7 +
4 X cubed.
We don't worry about the dividing by one. That's not
going to change anything.
Notice we don't multiply out the brackets. It's better to keep
the brackets there. We may want it in that form to work with
later. What if we
have something
like this? No obvious common factor,
but this is a
quadratic. And so because it's a quadratic expression, there is
the possibility of factorizing
it. If there's the possibility of Factorizing it, then what we
might find is that one of those factors could be X plus one, in
which case we could then divide top on bottom by that common
factor. So leave the top as it is and let's concentrate
on this bottom. We want to be able to factorize that.
That means two brackets.
An X in each bracket.
That ensures is the X squared.
We want to have.
2 as a result of multiplying these two numbers together that
go in here and here. So we'll have two and one.
Now we need a plus sign to give us the Plus 3X.
And what we can see is that X Plus one is a common factor,
so I'm going to put the brackets around that one on
top to show us. We've got a common factor and X plus one
into there goes 1X plus one into. There goes one and so
we're left with one over X +2.
You can have one that is if you like the other way up, so let's
have a look at that X minus 11 A plus 30. Sorry, A squared minus
11 A plus 30 over a minus 5.
Again, the A minus five. That's nice. That's OK. Let's keep that
together with a bracket, but let's have a look at this. A
squared minus eleven 8 + 30. Can we factorize it? Can we break it
down into two factors?
We've got a squared, so we need an A and and a. We now need 2
numbers that are going to multiply together to give us 30,
but add together to give us minus eleven. Well, six and five
seem a good bet for the 30, and if we make a minus six and minus
five, that ensures the plus 30 minus times by A minus. And it
also ensures the minus 11 a 'cause will have minus 6A and
minus 5A there.
We've now got a common factor of a minus five, so we can divide
the top by A minus five on the bottom by A minus five. So we
end up with just a minus 6, and that's our answer.
Now that's the proper way to do them. Some of you might be
tempted. Might be tempted when you see something like this.
Till forget what it is you're supposed to
do. So you might suddenly think are lots of threes. I can get
rid of some three, so let's cancel 3 here and a three there.
One problem. We said we had to cancel common factors.
3 doesn't appear here in this term, there's no factor of three
in this term. We cannot do this kind of canceling.
Let me just show you why not. If we have a look at a numerical
example. Supposing we had, let's say 5 + 3 over 3
+ 1. Now if we do the computation without doing
the canceling we get.
8 over 4.
And that's two, everybody is happy that that's the case.
But if I do what I did here and suddenly go absolutely bananas
and cancel the threes, then that's a one there and one
there. So what I seem to have now is 5 + 1, which is 6 over 1
+ 1, which is 2 gives me 3.
Two and three are not the same. Now we've agreed that
tools the correct answer, where have we gone wrong?
We cancel these threes divided by these threes, but three is
not a common factor because it doesn't appear in the Five and
it doesn't appear in the one.
So we can't do this over here by the same reasons what we have to
do is look and see if we can factorize what is on the top
three X squared plus 10X plus three. Can we factorize it?
Well, the X +3 is fine, let's keep it all together in a
bracket. Brackets here.
Three X squared will need a 3X and an X.
Will also need a three under one to make up this number here, and
we've got to get 10 out of it, so we're probably have to have
the three there to give us Nynex across there under one there.
Plus signs because these are all plus signs here and now. We can
see we have got this common factor of X plus three, so we
can divide the bottom by X +3 and the top by X +3.
That will give us 3X Plus One and that is the correct answer.
Sometimes we have to work again back a little bit harder, so
let's take this example 6X
cubed. Minus Seven X squared minus 5X all
over 2X plus one.
That doesn't give me much hope for is here, but there is a
common factor of X in each term, so perhaps we can take
that out as a common factor. To begin with. We might find
we can factorize what's left, so taking that X out as a
common factor is going to give us six X squared there.
Minus Seven X there.
And minus five there closed the bracket and we still to divide
by the 2X plus one. Also we hope. So now let's have a look
at this. 'cause this is now an
ordinary quadratics. X.
Six X squared? Well, let's have a guess at 3X and 2X.
After all, there's 2X down
there. What do we need now? We've got minus five to deal
with. I want if I can have it 2X plus one. So let's just be
guided by that. For the moment, let's make that 2X plus one.
Minus 5 is what I need to multiply by the one to give me
minus 5. Have I got it right? I need to check on that minus 7X
will hear I plus 3X and here I've minus 10X and that does
give me minus 7X. So that's right. Now I need to divide by
the 2X plus one. Let's get it together in a bracket.
I can divide top and bottom now by 2X plus one.
Goes into itself once and once
there. The answer is just what I'm left with here X times 3X
minus five, and again I leave it in its factorized form, not try
to multiply it out again 'cause I may need that form later on.
Supposing we take something like this
X cubed minus one over X minus one.
What now? Doesn't seem to be a common factor
in X cubed minus one.
Difficult. But you may know a factorization 4X cubed
minus one. It does in fact factorize as X minus
one brackets X squared.
Plus X plus one.
And so, because we know that factorization, we can see
straight away, we can simplify by dividing the
bottom by X minus one, and that goes in once and the top
by X minus one. And so we just left with the other
factor X squared plus X plus one.