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>> Math is cool and you can do it.
>> In this video we're going to learn some more methods of factoring trinomials
of the form ax squared + bx + c. So let's say you're factoring 6x squared - 11x + 4.
Right, so we've gone over how to do trial factors.
Well I'm going to ask you this question.
What if you knew one of the correct factors was 3x - 4?
So if you were sure that that was one of the factors could you figure
out what the other missing factor was?
Alright, so keep in mind I'm saying you definitely know 3x -
4 is the factor what would the other one be.
Well 3x times what will give you 6x squared?
2x. And -4 times what will give you 4?
-1. and then you can check this by doing the outer and inner term, doing the outer
and inner terms and making sure that really does add up to a negative 11x which it does,
negative 3x and negative 8x so that's your correct answer.
So wouldn't that be nice if you're always given at least one of the correct terms.
So let's see if you could do this one.
Factor 18m squared - 27 m + 4, if you know 6m - 1 is one factor, so try this.
You know 6m - 1 is one of the factors, so what would the other factor be?
So I'm trying to get 18m squared for my first term so that must be 3m and I'm trying
to get a positive 4 for my last term.
I have negative1 times what will give me that +4, well that'd be a -4 and then let's check
and make sure this really is the correct factorization.
We've already checked the first and last term, let's do the outer and inner.
We have negative 24m and negative 3m is negative 27m.
So this is the correct factorization of this one,
also the correct factorization for that one.
So the trick is if you can figure out a way of getting one of the correct factors it's not
so hard to get the next one, but you make sure you do always check.
Well I have a sneaky way that I'm going to teach you
which is something you should do a scratch work because if somebody looks
at your work they may not understand what you're doing to get one of the correct factors.
So here is how it goes.
Okay, here is my tricky method, what you're going to do is use the product and sum idea
but the product is not the last number here, the product is you have
to multiply the first coefficient times the last constant term that's the product.
So the product is 15 times 8, which is 120, okay.
So the product is 120 and the sum is still the middle coefficient.
So let's write down numbers that which I'm trying to find two numbers that have a product
of 120 and a sum of negative 22, which we've done before, so we've got 1 and 120, 2 and 60,
3 and 40, 4 and 30, 5, let's see if 5 goes into 30, 6 x 4 is 24, 6 20, 7 does not go into any
of those numbers, 8 does, 8 goes into 24, 3 times and so [inaudible] 15,
that's how it came from originally 15 times 8.
Let's see 9 doesn't go into it and 10 times 12, those are all the possibilities.
Sum is negative, right, so the bigger number has to be negative and the product is positive,
so the numbers have to have the same sign,
so these numbers are going to also be negative, alright.
So find two numbers that add up to negative 22 and there it is, alright.
Right, now what do we from here?
This is the sneaky part of the problem.
You're going to start off with the first original term in the problem, 15x squared,
so now you're going to write 15x squared and one of these numbers -- it doesn't matter which one,
negative 10x, so you're going to take the middle term which is negative 22x and instead
of writing the negative 22 that coefficient, you're going to write one of these numbers.
So I'm going to choose the negative 10x, can we move this over a little bit and change it
to another color so it's easy to see.
Okay, now you're going to factor these two terms.
So 15x squared - 10x by pulling out the greatest common factor and that's 5x times x -,
I'm sorry, that is 5x times (3x - 2).
Correct? Whatever you just put in the parentheses is one
of the correct factors, alright, so watch this.
Now you know it's (3x - 2), let's figure out the other one.
3x times what will give you 15x squared, 5x.
-2 times what will give me positive 8, negative 4, let's check it.
Does that give me the correct one, look at that, I've got -12x,
-10x that does give me the negative 22x, cool, and that's the sneaky way of using the product
and sum to get one of the correct factors, so I'm doing the product and sum,
one huge difference is the product is the first coefficient times the constant at the end.
You have to multiply those together.
So that's not just 8, it's 120.
I'll explain this later in another video why this works,
but let's just see how we can get the right answers for now.
Here's another one, factor 2x squared - 27x - 14.
Remember you don't have to use this method, you could use just do trial factors
and see if you get the correct answer.
Alright, so what I want to do is the product and sum again.
Okay, so the product, remember we're going to multiply the first times the last term
and that's 2 times -14, which is -28 and the sum is the middle term, which is -27,
but some of you might be able to come up with a correct answer right away here,
1 times 28 does it because if you put the negative in front of the bigger number
and use the opposite signs since it's negative, this is going to be it.
Alright, but let's say you didn't know that and you, you could always just list them all
and then at that point you could realize it is going to be 1 and the negative 28.
Alright, so now what's the next step?
You take the first term over here which is 2x squared, 2x squared and you take one
of these numbers, doesn't matter which one, I'm going to do the 1 + 1x,
so remember it's like you're writing this the first two terms, but you're putting
in the new number here, one of them instead of the negative 27x,
so you're only writing two terms and then you factor out the greatest common factor.
Alright, well its just an x, which is (2x + 1), that's one of the correct factors.
So we go over here, 2x + 1.
Alright, so what's the other factor?
Well, 2x times what will give me 2x squared,
x and 1 times what will give me negative 14, negative 14.
And that should be the correct factorization and you should check by going ahead
and doing the Foil method making sure
that you've got the correct inner and outer term which it does.
Now what if you had chosen, I said it didn't matter which one you chose,
whether you chose the +1 or the negative 28.
So instead what if you'd taken the first term 2x squared and done it with a -28x,
always remember to write the x. So you'll get a factor out at 2x and that gives me (x - 14),
that tells you that (x - 14) is one of the correct factors and then we have to figure
out the other one, x times what is 2x squared and -14 times what is -14, it's 1.
Well that's equivalent, both of those are exactly the same,
that's how to factor 2x squared - 27x - 14.
So this is what I call the sneaky way of finding one of the correct factors.
I'm going to stop this video now and you can go onto the next video, we'll do more examples.