[ Silence ]
>>All right we're going to go over how to recognize and factor perfect square trinomials.
All right so what is that?
Let's start off with a binomial a plus b, when you square it remember
that means a plus b times a plus b, so if you do the foil method and combine
like terms we get a squared plus 2ab plus b squared.
Similarly if instead it was a minus b the only difference is the middle term would be a
negative 2ab.
So summarizing we have that a plus b squared is a squared plus 2ab plus b squared
and a minus b squared is a squared minus 2ab plus b squared.
So what you notice here is if you squared something, a binomial, the first term happens
to be the first thing squared and the last term happens
to be the last thing squared, so the last term squared.
And both of them are positive, notice it says plus b squared so there must be a plus sign
at the end and it must be a perfect square.
On the other hand, look at the middle term.
The middle term in each of them has either a plus 2 times something
or a minus 2 times something.
If you have 2 times something, it's even, so the middle term must be even.
So first of all any time you note that the first and last term are perfect squares,
so first and last term must be perfect squares and that they have to be positives
since perfect squares are positive,
and the middle term must be even, it might be a perfect square.
Let's look at this example, 4x squared plus 12x plus 9.
Now you don't have to know that that's a perfect square or guess that it's a perfect square
or recognize it but actually this is a perfect square.
You could just go ahead and do the, the sum and product idea, trial factors
and eventually get the right answer.
So somebody with trial factors might, you know, try several things, that they'll try 2x and 2x
and they might try 9 and 1, doesn't work so they try 3 and 3, they do the first, inner, outer,
last and they say, oh, that's the right answer.
So when you're done you could leave this as the answer, 2x plus 3 squared
or you could write the answer as 2x plus 3, the whole thing is squared.
Remember that 2x plus 3 squared is not the same thing as 4x squared plus 9 okay?
What it means is 2x plus 3 times 2x plus 3.
So I did not recognize that was a perfect square.
In other words I just went through and hacked away and got the right answer.
But let's look at it again, 4x squared plus 12x plus 9.
Anytime you're given a trinomial, the first thing you might want to look
for is it may be a perfect square.
So I look at the first term, 4x squared and that's a perfect square
because it's 2x times 2x, I look at the last term, it's positive
and it's 9 which is also a perfect square.
All right, so the first and last term are perfect squares, the middle term is even,
this is a clue that maybe it's a perfect square.
So what we do is we try that for our trial factors first.
That's how I really think of it.
So if it were a perfect square, the first term would have to be what so that
when I squared I get 4x squared, it would have to be 2x.
And the last term would have to be what to get 9, 3 or negative 3.
Whatever the middle sign is, that's the sign of the problem so this is plus.
So if it were a perfect square, it would have to be the 2x plus 3
to get the correct first and last term.
But what you need to check is the middle term.
The middle term is always 2 times each of these two terms here.
So you're going to check, 2 times 2x times 3, does that equal 12x?
The middle term, and the answer is yes so then I know for sure
that was the correct factorization.
So it's a shortcut for trying a lot of trial factors, that would be the first one you tried.
All right, look at this one, 4x squared plus 13x plus 9.
So you say well, the first term's a perfect square, the last term's a perfect square.
Is the middle term even?
No. Well this one's not a perfect square so then you go and do your regular, oh,
the trial factors or try the sum and product or the box method and this factor's not
as a perfect square but this one happens to be 4x times x and I think this is 9
and 1 and it's not a perfect square.
I did it with trial factors and came up with it on my first try only because I've done thousands
of these so don't, don't be upset with yourself if you can't get it that quickly
but the first thing I would do is just sort of do a quick check
to see if it might be a perfect square.
Okay here's another problem, I look at it and I notice the first term is a perfect square
and the last term's a perfect square and the middle term's even so what I'm hoping is
that this is a perfect square and I'll get it on my first try.
So is this just one thing squared?
Well the middle term is negative so first of all I know I have a minus sign in the middle.
All right, so let's see, what times itself gives you 16x squared?
Four x. And what times itself gives you 9?
Three. All right, so if it were a perfect square it would have to be 4x minus 3
to get the correct first and last term.
So now we've got to just check that middle term, it's always 2 times each
of these in here, 2 times 4x times 3.
I'm just looking at the number part not worrying about whether it's positive or negative
because I've already taken care of that issue and let's see that is 24x, so yes, this is a,
this is the correct answer and you're done.
So it's a shortcut to getting the right answer more quickly.
Here's another problem, we notice the first term is a perfect square,
x squared and we notice the last term is negative 16.
Negative 16 is not a perfect square, positive 16 is a perfect square
so this cannot be a perfect square.
So we do the sum and product or trial factors but it is not going to be a perfect square
because of this negative sign, even though the middle term is even,
even though the first term is a perfect square.
All conditions have to be met to even have it be a possibility.
So this is x and x, in this case I need two numbers that multiply out to be 16 add up to 6,
I think that's going to be a plus 8 and minus 2.
So this one does not happen to be a perfect square.
So what you have to do, this is all about recognizing something
that might be a perfect square and it's a shortcut to getting the correct factorization
without trying a lot of other things first.
All right let's look at this problem.
The first step in factoring before you even think about difference of two squares,
perfect squares is to look for the greatest common factor, any common factor.
In this case there is a common factor, do you see what it is?
It's m cubed, so we take out the m cubed and that gives us m squared plus 18m plus 81.
And now when I look inside my parentheses, well, the first term is a perfect square,
the last term's a perfect square and the middle term's even.
So I'm going to hope this is a perfect square.
[ silence ]
Well if were a perfect square, all right, of a binomial,
first of all they're all positive so the middle term's positive.
What times itself is m squared?
M. And what times itself is 81?
Nine. And let's just check the middle term.
The middle term is always 2 times the m times the 9, so 2 times the m times the 9 is 18m,
that was the middle term, so yes there's the correct factorization.
Sometimes if you want to remember what to multiply by, this 2 times m times 9,
I look at the 2 and the 9 and the m, everything you see you just multiply together,
don't square it but 2 times 9 times m, or m times 9 times 2.
So that's true on the previous problems too.
See if I check this one if you look at your answer you're seeing 2x, a 3 and a 2,
just multiply them altogether and see if it's 12x.
Here's another one to check the middle term.
You're doing the 4x times the 3 times the 2
or you could say you're doing the 4x times the negative 3 times the 2 if you want
to check the sign, make sure it's negative as well.
So the key to this is recognizing that something might be a perfect square and trying it,
but the key is especially that you need to check your answer, you have to multiply out.
Sometimes you're given something that looks like it might factor,
it looks like it might be a perfect square like x
to the 4th plus 6x squared plus 1, but it's not.
You could try it and it doesn't work if you did the foil method for instance.
And in fact, this one happens to be prime, no matter what you try.
Because that's the only thing you can try, in this case it doesn't work.
Okay? That's it.