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