Introduction to Factoring Polynomials

So the first and most important thing you want to do when you're factoring is you want to look for the greatest common factor. So this is something that you can divide out of all the terms and that's the first step. You're going to want to do that. That's going to make all the numbers smaller. It's going to make it easier to go ahead and factor using some of the other techniques because the numbers are going to be smaller. So once you factor out the greatest common factor, if it does have one, it might not have one, then you have to ask yourself does the polynomial have two terms, three terms, four terms, or maybe even more. So if it has two terms, it could be a difference of two squares. Okay, so you can see a squared minus b squared, so a perfect square minus a perfect square. Or it could be a difference or sum of two cubes. So you have to see if they're both perfect cubes, either added together or subtracted. Okay, and then we have a formula for factoring those. Or does it have four terms? Okay. In that case you would want to try factoring by grouping. So grouping, you're going to be putting parentheses around the first two terms and the last two terms. I'll be showing you examples of all these different techniques. And then if there's three terms, you want to ask yourself, is the leading coefficient one? The number in front of the x squared, is it one? Or is it something other than one? And you can see here I've written this as a. And the technique I'm going to be showing you is how to split the middle term. then go ahead and factor by grouping so you'll have four terms at that point. So let's go through the different types. Go ahead and pause the video if you want to try it on your own and then you know see if you did it right or if you need to repeat the video go ahead and do so. So let's start off with the first type the greatest common factor. So I've written some down here. So let's try this first one. 5x squared plus 10x plus 20. So we've got 5x squared plus 10x plus 20. Okay so The terms are separated by plus or minus. So you can see this is a three terms trinomial. And it looks like all the terms are divisible by five. So what we're going to do is you can do this. You can actually go ahead and... Draw a division bar, okay, a fraction sign, and divide them all by 5, and you can see you're going to be left with x squared plus 2x plus 4. But what happens to that 5 is that it's going to be here in front of the parentheses. So it's like doing the distributive property backwards. If you want to check your work, all you have to do is distribute the 5 back into the parentheses, okay, and you're going to get back the original polynomial that we started with, 5x squared plus 10x plus 20. Okay, let's look at another example. Let's do the next one here. Let's do 6x cubed minus 2xy squared. Okay, so now for this example, this is a binomial, two terms separated by this subtraction sign, okay, the two groups. What can you divide out of both of these terms? Well, it looks like with the 6 and the 2, those are both divisible by 2, okay? And it looks like there's one x in common, so we can divide out an x out of both these groups. And it looks like that's all that we have in common. So what we're going to do is we're going to divide, and we've got 3x squared minus, okay, the 2's cancel, the x's cancel, we're left with y squared. And now what happens to the 2x? That's going to come out in front of the parentheses. okay, like so. And if you distribute back in, okay, if you distribute the 2x back into the parentheses using the distributive property, you're going to get back the original polynomial that we started with. So that's the first step in factoring. You don't want to forget this. Sometimes students will skip over the greatest common factor. They'll want to get right into some of the other techniques depending on how many terms there are. But if you do this first, it's going to make your work easier as you go on in the factoring process. But you want to Factoring is like reducing fractions. You keep reducing, reducing, reducing. Well, factoring, you keep factoring, factoring, factoring until you can't factor any further. So let's get into the next type of factoring. So I'm just going to erase this real quick here for us. Okay, the next type of factoring is looking at how many terms. Do you have two terms, three terms, or four terms? So let's look at some examples where we have two terms. So I wrote down some examples for us. Let's look at this one here, x squared minus 100. Okay so what we want to ask ourselves, okay it has two terms, you can see it's the terms are separated by the minus sign, but is it a difference of two squares? Is it a difference of two cubes? Or is it a sum of two cubes? Well you can see x squared is a perfect square and a hundred is also a perfect square because 10 times 10. So what we're going to do, this is a difference of two squares, you can see a squared minus b squared it factors as a plus b times a minus b. So in this case our a value is the square root of x squared which is x. Our b value is 10 because the square root of 100 is 10 and we're going to make one of these plus and one of them minus. It doesn't matter the order you could make this minus and this plus. Now if you want to check your work all you have to do is FOIL these together so you can distribute the x and then distribute the 10 and you're going to get back the original x squared minus 100. The reason is, is 10x and negative 10x, those cancel out. That's why we don't have a middle term. Let's look at another example of two terms. Let's look at this one, 4y squared minus 100. Okay, now remember with factoring, most important step is the greatest common factor. So we want to do that first. Is there anything that we can divide out of both of these terms? That's right, we can divide out a 4. So we're going to factor out the 4. And then we're going to look at what's left and say, can we factor that further? Well, you can see this is a difference of two perfect squares, okay, difference of two squares. So we factor that as a plus b, a minus b. So that's going to be y plus 5, okay, because the square root of 25 is 5, square root of y squared is y, and y minus 5. Sometimes this is called the sum and difference pattern. What happens to the 4? You just bring that down in front. So the idea is if you were to multiply all these groups together, you're going to get back the original polynomial. And you can go ahead and do that just to make sure you did the right factoring. Okay, so that's working with difference of two squares. Let's see if we can look at difference in sum of two cubes now. I'll show you some examples for those. Now remember, you always want to check for the greatest common factor first, but let's do this one, x cubed minus 8. Okay, so the first thing you want to do is check for the greatest common factor. It doesn't look like there's anything that we can divide out of both of these terms. And we can see there's two terms, so it either has to be a difference of two squares, or a difference of two cubes, or sum of two cubes, or it could be neither. But it looks like this is a difference of two cubes, because 8 is 2 cubed, and you can see x is cubed, and we're subtracting, so we've got a difference of two cubes. And you can see here, I combined these formulas together, but when you're adding two cubes, cubes you add, subtract, and add. When you subtract two cubes you subtract, add, and add. Some students they'll remember the acronym SOAP. You can think of it as same, opposite, always positive. So same meaning if you're adding you add then it's the opposite for the second okay sign and then the last one is always positive. So let's look at this one here. So here you can see X is A and and two is b, okay, in this formula here. So what we're gonna do, it's x minus two, x squared plus two x plus two squared, which is four. And we can't factor this any further, so we're done. Let's look at another example. Let's do this one here. Two x cubed minus 54. Let's do plus 54. So here again the first step is, is there a greatest common factor that we can factor out? If so, do that first. Looks like both of these terms are divisible by 2. Okay, so what we're going to do is we're going to factor out the 2. We've got x cubed plus 54 divided by 2 is 27. And then you ask yourself, can we factor it further? Well, 27 is a perfect cube. It's 3 cubed. And x is a perfect cube. It's x cubed. So I'm just rewriting this, okay, just in a different form. So now we can see, all right, it's a sum of two cubes. So let's go ahead and use our sum of two cubes formula. And so we're going to do that now. So this is going to be x plus 3. It's the same sign because we're adding or adding. x squared minus the opposite sign, a times b, which is 3 times x. plus 3 squared 3 squared is 3 times 3 9 and The only other thing you have to do is bring down the 2 if you multiply all these together You're going to get back the original 2x cubed plus 54 Okay, so that's working with When you have two terms, it can be a difference of two squares or a difference of two cubes or sum of two cubes. There isn't a sum of two squares at this level of factoring where you're working with real numbers. Okay, so then the next one we want to talk about is factoring by group. Okay, so what is factoring by grouping? Well, I'll show you. So when you have four terms, you want to try this technique. And again, remember the terms are separated by adding, okay, or subtracting by plus or minus. So say, for example, we want to factor. This one right here, x cubed minus 2x squared minus 4x plus 8. Okay, so notice you have 1, 2, 3, 4 groups separated by minus or plus, and we want to factor by grouping. We're going to try that technique because we have four terms. Okay, so the first thing I'm going to do is I'm going to group the first two. and subtraction is like adding the opposite so I'm just going to do that and I'm going to group the last two okay then what we're going to do is we're going to factor out the greatest common factor out of this group and out of this group so it looks like we can factor out an x squared out of both of these terms so we have x minus 2 and it looks like we can factor out a negative 4 out of both of these terms so we're left with x minus 2 now what we have we have two groups we have this group and we have this group Notice how they both have an x minus 2. See this x minus 2 in both groups? What we're going to do now is we're going to factor out the x minus 2 as a greatest common factor a second time, right? So we're going to factor that out. So it's like dividing this group by x minus 2. It's like dividing this group by x minus 2. When you do that, these are going to cancel. You're going to be left with x squared. These are going to cancel. You're going to be left with minus 4. What happens to the x minus 2, though? That comes out in front like so. So if you were to take this x minus 2, distribute it to the x squared, you'd get back this group. And if you take the x minus 2 as a whole and distribute it to the negative 4, you'd get back this group here. Okay, now factoring is one of those... things that you have to keep seeing if you can factor further. Okay, you want to see if you can keep breaking it down further. Look what we have here. We have a difference of two squares. Okay, so we're back to here, and we're going to use our sum and difference pattern. So this is going to be factored to x plus 2 and x minus 2, and then we're going to bring down the x minus 2. We can combine these, since we have two x minus 2s, into x minus 2 squared, and so there's going to be your final factored form. So this is factoring by grouping when you have four terms. Okay, so now let's talk about when you have three terms. Okay, so three terms. So let me show you an example for this one. Make a little room for us here. Now there's two different types here with the three terms. Okay, there's the type where you have a leading coefficient of one. Okay, see one in front of the x squared. And then there's a type where you have something other than 1, like a 2 or a negative 3, something like that, like an a value other than 1. So we're going to look at both these types. Let's look at this one first. This one's a little bit easier when the leading coefficient's 1. And the example I've written down here for us is x squared plus 7x plus 12. Okay, so you can see three terms. Now, when the leading coefficient, see, it's 1, you just ask yourself, what two numbers multiply to this last term, positive 12? Okay, I'm sorry, I made a mistake. They multiply to the last term, 12, but they add to the middle term, 7. Okay, I just want to make sure I said that right. So multiply to 12, but add to 7. Okay, so that's very easy. We just say, hmm, it could be 1 and 12, 2 times 6, 3 times 4. 3 times 4 is 12. 3 plus 4 is 7. So those are the ones we want. 3 and 4. 4 and again you can check your work by distributing and then distributing the 3 as well and you'll get back the original okay because you can see 3x plus 4x is 7x okay 3 times 4 gives you back the 12 and the x times x gives you x squared so again just remember what multiplies to 12 But adds to the middle coefficient 7. Let's do another example like that so you can see. Say we have this one x squared minus 5x minus 24. Okay, you can see it's a trinomial, three terms. What two numbers multiply to negative 24? but they add to negative 5? Well, it's a good question. We know one of them has to be positive and one has to be negative when you multiply them to get negative 24. So we'll do plus and minus. And we could do 1 times 24, 2 times 12, 3 times 8, 4 times 6. But you can see this is going to be 3 and 8, okay, because 3x and negative 8x add up to our middle term, negative 5x. Now, again, you want to see if you can factor further. Okay, but in this case, this is as far as we can go, so this is the fully factored form. You also want to see if there's something you can factor out first. Okay, the greatest common factor, but in this case, there wasn't something that we could divide out of all the terms, so we went right into factoring when you have three terms, leading coefficient of one. Okay, let's get into the other type of trinomial that we want to factor, and that's where the leading coefficient is something other than one. Now, sometimes, Sometimes, no matter how much you try, okay, or how much you want to be able to factor something, it just might not be able to be factored. And in that case, they call it prime, okay, just like a prime number. Like if you have the number 7, you know, it's only divisible by 1 in itself. And it's the same thing with these. Sometimes you just can't factor them. They're considered prime. So let's look at some examples here. I've written down two for us. We've got 6x squared minus 11x minus 10. Okay, so what we're going to do in this problem here is first thing, you look to see if there's a greatest common factor. Doesn't look like there's anything we can divide out of all three terms. Okay, we know it's three terms, and we know the leading coefficient is something other than one. So what we're going to do is we're going to split the middle term, and then we're going to factor by grouping. So here's the technique. You're going to take the leading coefficient times the constant. So we're going to take 6 times negative 10, okay, which comes out to negative 60. And then we're going to solve this problem here. We're going to say what two numbers multiply to negative 60, but they add to the middle coefficient, negative 11. Okay, do you follow? So we want to say what two numbers multiply to the product of the leading coefficient and the constant, negative 60. but they have to add to the middle coefficient negative 11. Well there's a lot of possibilities you could do 1 times negative 60 or 2 times negative 30 but it looks like for this example it's going to be negative 15 and positive 4 because negative 15 plus 4 is negative 11 and negative 15 times 4 is negative 60. We're going to use these two numbers to help us split. the middle term into two groups. We're going to write it as negative 15x and positive 4x. We're going to bring down the negative 10 and we're going to bring down the 6x squared. Now some students will ask me this all the time. They say, does it matter if I put negative 15x here or here? You know, does it matter the order? And the order doesn't matter. The main thing is you can see that negative 15x and positive 4x add up to that middle term negative 11x. So we haven't changed the problem. We've just divided it into two separate groups. So what we're going to do now is we're going to go to the 1, 2, 3, 4, 4 terms. We're going to try factoring by grouping. Okay, and so what we're going to do is we're going to group these like so. We're going to group these like so. We're going to factor out the greatest common factor, which in this case is 3x. And here the greatest common factor is 2. And if all goes well, you can see that we have a 2x minus 5. We have the same quantity here in common between this group and this group. So we're going to factor out the 2x minus 5, okay, as if we were doing the distributive property backwards, right? We're dividing out the 2x minus 5. And so what we're going to be left with, okay, is 3x plus 2. Okay, this part can be a little bit confusing sometimes for students, but imagine if you're dividing this by 2x minus 5, what are you left with? 3x. If you divide this by 2x minus 5, you're left with 2. That 2x minus 5 doesn't go away, it's just you're dividing it out. You're factoring it out. If you were to take this as a whole, distribute it to 3x, you'd be left with this first group here. If you take the 2x minus 5, distribute this whole group to the 2, you'd be left with that group there. Now, you want to check to see if you can factor it further, but it looks like this is as far as we're going to be able to factor this. Okay, it doesn't look like there's anything else that can be done, so it's completely factored. Let's look at one more example like that. And then... We'll just kind of conclude this video. This is a beginner to intermediate level. There's more advanced types of problems that we can factor, but if you can master these steps here, you'll be well positioned to do the other steps, the more advanced steps as you get to them. So let's look at this last problem here. We've got 10x squared minus 3x minus 4. Okay, so we see it's a trinomial, three terms. We see if there's a greatest common factor. That's the first step, but there isn't. We can see the leading coefficient is something other than 1. So we're going to look at splitting the middle term and factoring by grouping. So the first step is to multiply 10 times negative 4. So we say what two numbers multiply to negative 40, but they have to add to negative 3. Okay, so what would that be? Multiply to negative 40, the product of the leading coefficient and the constant, but they have to add to the middle coefficient, which is negative 3. Well, we could do 1 and negative 40, 2 and negative 20, 4 and negative 10. We go through the different... combinations, but they also have to at the same time add to negative 3. So for this one it's going to be negative 8 and positive 5. That multiplies to negative 40 but adds to negative 3 and that helps us to split this middle term into negative 8x plus 5x. Okay so see how those add up to the middle term negative 3x. We're going to bring down the 10x squared. We're going to bring down the minus 4. Now you can see we have one, two, three, four terms. Again, the terms are separated by minus or plus. We have four terms. That's when we do the factoring by grouping, okay? So what we're going to do is we're going to group the first two by putting parentheses around them. We're going to group the last two, okay? Now this isn't factored because see how there's a plus sign in between the two? It's not like this times this equals this. These are added. So what we're going to do is we're going to factor out the greatest common factor. Here it looks like we can factor out a 2x. So all I did was divide this by 2x, which leaves us 5x. I divided this by 2x, which leaves us negative 4. This one, there's not a greatest common factor. So really, you could write this as plus 1. You're just factoring out 1. 1 times anything is itself, right? Now you can see there's a 5x minus 4 in both groups. We're going to factor out or divide out that 5x minus 4. And so what we have is 5x minus 4 times 2x plus 1. Okay, just again really quick. So if you divide this by 5x minus 4, you're left with 2x. If you divide this by 5x minus 4, you're left with 1. You can check your work. If you distribute the 5x minus 4 as a whole, as a group, to the 2x, you get back this term. If you distribute this all to 1, you get back this term. So you can check to see if you can factor any further, but it looks like that's as far as we can take this one.

It's going to make it easier to go ahead and factor using some of the other techniques because the numbers are going to be smaller. So once you factor out the greatest common factor, if it does have one, it might not have one, then you have to ask yourself does the polynomial have two terms, three terms, four terms, or maybe even more. So if it has two terms, it could be a difference of two squares.

Okay, so you can see a squared minus b squared, so a perfect square minus a perfect square. Or it could be a difference or sum of two cubes. So you have to see if they're both perfect cubes, either added together or subtracted.

Okay, and then we have a formula for factoring those. Or does it have four terms? Okay. In that case you would want to try factoring by grouping.

So grouping, you're going to be putting parentheses around the first two terms and the last two terms. I'll be showing you examples of all these different techniques. And then if there's three terms, you want to ask yourself, is the leading coefficient one?

The number in front of the x squared, is it one? Or is it something other than one? And you can see here I've written this as a. And the technique I'm going to be showing you is how to split the middle term.

then go ahead and factor by grouping so you'll have four terms at that point. So let's go through the different types. Go ahead and pause the video if you want to try it on your own and then you know see if you did it right or if you need to repeat the video go ahead and do so. So let's start off with the first type the greatest common factor.

So I've written some down here. So let's try this first one. 5x squared plus 10x plus 20. So we've got 5x squared plus 10x plus 20. Okay so The terms are separated by plus or minus.

So you can see this is a three terms trinomial. And it looks like all the terms are divisible by five. So what we're going to do is you can do this. You can actually go ahead and...

Draw a division bar, okay, a fraction sign, and divide them all by 5, and you can see you're going to be left with x squared plus 2x plus 4. But what happens to that 5 is that it's going to be here in front of the parentheses. So it's like doing the distributive property backwards. If you want to check your work, all you have to do is distribute the 5 back into the parentheses, okay, and you're going to get back the original polynomial that we started with, 5x squared plus 10x plus 20. Okay, let's look at another example.

Let's do the next one here. Let's do 6x cubed minus 2xy squared. Okay, so now for this example, this is a binomial, two terms separated by this subtraction sign, okay, the two groups. What can you divide out of both of these terms? Well, it looks like with the 6 and the 2, those are both divisible by 2, okay?

And it looks like there's one x in common, so we can divide out an x out of both these groups. And it looks like that's all that we have in common. So what we're going to do is we're going to divide, and we've got 3x squared minus, okay, the 2's cancel, the x's cancel, we're left with y squared. And now what happens to the 2x?

That's going to come out in front of the parentheses. okay, like so. And if you distribute back in, okay, if you distribute the 2x back into the parentheses using the distributive property, you're going to get back the original polynomial that we started with. So that's the first step in factoring. You don't want to forget this.

Sometimes students will skip over the greatest common factor. They'll want to get right into some of the other techniques depending on how many terms there are. But if you do this first, it's going to make your work easier as you go on in the factoring process. But you want to Factoring is like reducing fractions. You keep reducing, reducing, reducing.

Well, factoring, you keep factoring, factoring, factoring until you can't factor any further. So let's get into the next type of factoring. So I'm just going to erase this real quick here for us. Okay, the next type of factoring is looking at how many terms.

Do you have two terms, three terms, or four terms? So let's look at some examples where we have two terms. So I wrote down some examples for us. Let's look at this one here, x squared minus 100. Okay so what we want to ask ourselves, okay it has two terms, you can see it's the terms are separated by the minus sign, but is it a difference of two squares? Is it a difference of two cubes?

Or is it a sum of two cubes? Well you can see x squared is a perfect square and a hundred is also a perfect square because 10 times 10. So what we're going to do, this is a difference of two squares, you can see a squared minus b squared it factors as a plus b times a minus b. So in this case our a value is the square root of x squared which is x. Our b value is 10 because the square root of 100 is 10 and we're going to make one of these plus and one of them minus. It doesn't matter the order you could make this minus and this plus.

Now if you want to check your work all you have to do is FOIL these together so you can distribute the x and then distribute the 10 and you're going to get back the original x squared minus 100. The reason is, is 10x and negative 10x, those cancel out. That's why we don't have a middle term. Let's look at another example of two terms. Let's look at this one, 4y squared minus 100. Okay, now remember with factoring, most important step is the greatest common factor.

So we want to do that first. Is there anything that we can divide out of both of these terms? That's right, we can divide out a 4. So we're going to factor out the 4. And then we're going to look at what's left and say, can we factor that further? Well, you can see this is a difference of two perfect squares, okay, difference of two squares.

So we factor that as a plus b, a minus b. So that's going to be y plus 5, okay, because the square root of 25 is 5, square root of y squared is y, and y minus 5. Sometimes this is called the sum and difference pattern. What happens to the 4?

You just bring that down in front. So the idea is if you were to multiply all these groups together, you're going to get back the original polynomial. And you can go ahead and do that just to make sure you did the right factoring.

Okay, so that's working with difference of two squares. Let's see if we can look at difference in sum of two cubes now. I'll show you some examples for those.

Now remember, you always want to check for the greatest common factor first, but let's do this one, x cubed minus 8. Okay, so the first thing you want to do is check for the greatest common factor. It doesn't look like there's anything that we can divide out of both of these terms. And we can see there's two terms, so it either has to be a difference of two squares, or a difference of two cubes, or sum of two cubes, or it could be neither. But it looks like this is a difference of two cubes, because 8 is 2 cubed, and you can see x is cubed, and we're subtracting, so we've got a difference of two cubes.

And you can see here, I combined these formulas together, but when you're adding two cubes, cubes you add, subtract, and add. When you subtract two cubes you subtract, add, and add. Some students they'll remember the acronym SOAP. You can think of it as same, opposite, always positive. So same meaning if you're adding you add then it's the opposite for the second okay sign and then the last one is always positive.

So let's look at this one here. So here you can see X is A and and two is b, okay, in this formula here. So what we're gonna do, it's x minus two, x squared plus two x plus two squared, which is four. And we can't factor this any further, so we're done.

Let's look at another example. Let's do this one here. Two x cubed minus 54. Let's do plus 54. So here again the first step is, is there a greatest common factor that we can factor out?

If so, do that first. Looks like both of these terms are divisible by 2. Okay, so what we're going to do is we're going to factor out the 2. We've got x cubed plus 54 divided by 2 is 27. And then you ask yourself, can we factor it further? Well, 27 is a perfect cube.

It's 3 cubed. And x is a perfect cube. It's x cubed. So I'm just rewriting this, okay, just in a different form. So now we can see, all right, it's a sum of two cubes.

So let's go ahead and use our sum of two cubes formula. And so we're going to do that now. So this is going to be x plus 3. It's the same sign because we're adding or adding.

x squared minus the opposite sign, a times b, which is 3 times x. plus 3 squared 3 squared is 3 times 3 9 and The only other thing you have to do is bring down the 2 if you multiply all these together You're going to get back the original 2x cubed plus 54 Okay, so that's working with When you have two terms, it can be a difference of two squares or a difference of two cubes or sum of two cubes. There isn't a sum of two squares at this level of factoring where you're working with real numbers. Okay, so then the next one we want to talk about is factoring by group.

Okay, so what is factoring by grouping? Well, I'll show you. So when you have four terms, you want to try this technique.

And again, remember the terms are separated by adding, okay, or subtracting by plus or minus. So say, for example, we want to factor. This one right here, x cubed minus 2x squared minus 4x plus 8. Okay, so notice you have 1, 2, 3, 4 groups separated by minus or plus, and we want to factor by grouping.

We're going to try that technique because we have four terms. Okay, so the first thing I'm going to do is I'm going to group the first two. and subtraction is like adding the opposite so I'm just going to do that and I'm going to group the last two okay then what we're going to do is we're going to factor out the greatest common factor out of this group and out of this group so it looks like we can factor out an x squared out of both of these terms so we have x minus 2 and it looks like we can factor out a negative 4 out of both of these terms so we're left with x minus 2 now what we have we have two groups we have this group and we have this group Notice how they both have an x minus 2. See this x minus 2 in both groups?

What we're going to do now is we're going to factor out the x minus 2 as a greatest common factor a second time, right? So we're going to factor that out. So it's like dividing this group by x minus 2. It's like dividing this group by x minus 2. When you do that, these are going to cancel.

You're going to be left with x squared. These are going to cancel. You're going to be left with minus 4. What happens to the x minus 2, though?

That comes out in front like so. So if you were to take this x minus 2, distribute it to the x squared, you'd get back this group. And if you take the x minus 2 as a whole and distribute it to the negative 4, you'd get back this group here. Okay, now factoring is one of those... things that you have to keep seeing if you can factor further.

Okay, you want to see if you can keep breaking it down further. Look what we have here. We have a difference of two squares. Okay, so we're back to here, and we're going to use our sum and difference pattern. So this is going to be factored to x plus 2 and x minus 2, and then we're going to bring down the x minus 2. We can combine these, since we have two x minus 2s, into x minus 2 squared, and so there's going to be your final factored form.

So this is factoring by grouping when you have four terms. Okay, so now let's talk about when you have three terms. Okay, so three terms. So let me show you an example for this one. Make a little room for us here.

Now there's two different types here with the three terms. Okay, there's the type where you have a leading coefficient of one. Okay, see one in front of the x squared.

And then there's a type where you have something other than 1, like a 2 or a negative 3, something like that, like an a value other than 1. So we're going to look at both these types. Let's look at this one first. This one's a little bit easier when the leading coefficient's 1. And the example I've written down here for us is x squared plus 7x plus 12. Okay, so you can see three terms.

Now, when the leading coefficient, see, it's 1, you just ask yourself, what two numbers multiply to this last term, positive 12? Okay, I'm sorry, I made a mistake. They multiply to the last term, 12, but they add to the middle term, 7. Okay, I just want to make sure I said that right.

So multiply to 12, but add to 7. Okay, so that's very easy. We just say, hmm, it could be 1 and 12, 2 times 6, 3 times 4. 3 times 4 is 12. 3 plus 4 is 7. So those are the ones we want. 3 and 4. 4 and again you can check your work by distributing and then distributing the 3 as well and you'll get back the original okay because you can see 3x plus 4x is 7x okay 3 times 4 gives you back the 12 and the x times x gives you x squared so again just remember what multiplies to 12 But adds to the middle coefficient 7. Let's do another example like that so you can see. Say we have this one x squared minus 5x minus 24. Okay, you can see it's a trinomial, three terms.

What two numbers multiply to negative 24? but they add to negative 5? Well, it's a good question. We know one of them has to be positive and one has to be negative when you multiply them to get negative 24. So we'll do plus and minus.

And we could do 1 times 24, 2 times 12, 3 times 8, 4 times 6. But you can see this is going to be 3 and 8, okay, because 3x and negative 8x add up to our middle term, negative 5x. Now, again, you want to see if you can factor further. Okay, but in this case, this is as far as we can go, so this is the fully factored form. You also want to see if there's something you can factor out first. Okay, the greatest common factor, but in this case, there wasn't something that we could divide out of all the terms, so we went right into factoring when you have three terms, leading coefficient of one.

Okay, let's get into the other type of trinomial that we want to factor, and that's where the leading coefficient is something other than one. Now, sometimes, Sometimes, no matter how much you try, okay, or how much you want to be able to factor something, it just might not be able to be factored. And in that case, they call it prime, okay, just like a prime number. Like if you have the number 7, you know, it's only divisible by 1 in itself.

And it's the same thing with these. Sometimes you just can't factor them. They're considered prime. So let's look at some examples here. I've written down two for us.

We've got 6x squared minus 11x minus 10. Okay, so what we're going to do in this problem here is first thing, you look to see if there's a greatest common factor. Doesn't look like there's anything we can divide out of all three terms. Okay, we know it's three terms, and we know the leading coefficient is something other than one.

So what we're going to do is we're going to split the middle term, and then we're going to factor by grouping. So here's the technique. You're going to take the leading coefficient times the constant. So we're going to take 6 times negative 10, okay, which comes out to negative 60. And then we're going to solve this problem here.

We're going to say what two numbers multiply to negative 60, but they add to the middle coefficient, negative 11. Okay, do you follow? So we want to say what two numbers multiply to the product of the leading coefficient and the constant, negative 60. but they have to add to the middle coefficient negative 11. Well there's a lot of possibilities you could do 1 times negative 60 or 2 times negative 30 but it looks like for this example it's going to be negative 15 and positive 4 because negative 15 plus 4 is negative 11 and negative 15 times 4 is negative 60. We're going to use these two numbers to help us split. the middle term into two groups.

We're going to write it as negative 15x and positive 4x. We're going to bring down the negative 10 and we're going to bring down the 6x squared. Now some students will ask me this all the time. They say, does it matter if I put negative 15x here or here? You know, does it matter the order?

And the order doesn't matter. The main thing is you can see that negative 15x and positive 4x add up to that middle term negative 11x. So we haven't changed the problem. We've just divided it into two separate groups. So what we're going to do now is we're going to go to the 1, 2, 3, 4, 4 terms.

We're going to try factoring by grouping. Okay, and so what we're going to do is we're going to group these like so. We're going to group these like so. We're going to factor out the greatest common factor, which in this case is 3x. And here the greatest common factor is 2. And if all goes well, you can see that we have a 2x minus 5. We have the same quantity here in common between this group and this group.

So we're going to factor out the 2x minus 5, okay, as if we were doing the distributive property backwards, right? We're dividing out the 2x minus 5. And so what we're going to be left with, okay, is 3x plus 2. Okay, this part can be a little bit confusing sometimes for students, but imagine if you're dividing this by 2x minus 5, what are you left with? 3x.

If you divide this by 2x minus 5, you're left with 2. That 2x minus 5 doesn't go away, it's just you're dividing it out. You're factoring it out. If you were to take this as a whole, distribute it to 3x, you'd be left with this first group here.

If you take the 2x minus 5, distribute this whole group to the 2, you'd be left with that group there. Now, you want to check to see if you can factor it further, but it looks like this is as far as we're going to be able to factor this. Okay, it doesn't look like there's anything else that can be done, so it's completely factored.

Let's look at one more example like that. And then... We'll just kind of conclude this video. This is a beginner to intermediate level.

There's more advanced types of problems that we can factor, but if you can master these steps here, you'll be well positioned to do the other steps, the more advanced steps as you get to them. So let's look at this last problem here. We've got 10x squared minus 3x minus 4. Okay, so we see it's a trinomial, three terms.

We see if there's a greatest common factor. That's the first step, but there isn't. We can see the leading coefficient is something other than 1. So we're going to look at splitting the middle term and factoring by grouping.

So the first step is to multiply 10 times negative 4. So we say what two numbers multiply to negative 40, but they have to add to negative 3. Okay, so what would that be? Multiply to negative 40, the product of the leading coefficient and the constant, but they have to add to the middle coefficient, which is negative 3. Well, we could do 1 and negative 40, 2 and negative 20, 4 and negative 10. We go through the different... combinations, but they also have to at the same time add to negative 3. So for this one it's going to be negative 8 and positive 5. That multiplies to negative 40 but adds to negative 3 and that helps us to split this middle term into negative 8x plus 5x. Okay so see how those add up to the middle term negative 3x.

We're going to bring down the 10x squared. We're going to bring down the minus 4. Now you can see we have one, two, three, four terms. Again, the terms are separated by minus or plus. We have four terms.

That's when we do the factoring by grouping, okay? So what we're going to do is we're going to group the first two by putting parentheses around them. We're going to group the last two, okay?

Now this isn't factored because see how there's a plus sign in between the two? It's not like this times this equals this. These are added. So what we're going to do is we're going to factor out the greatest common factor.

Here it looks like we can factor out a 2x. So all I did was divide this by 2x, which leaves us 5x. I divided this by 2x, which leaves us negative 4. This one, there's not a greatest common factor.

So really, you could write this as plus 1. You're just factoring out 1. 1 times anything is itself, right? Now you can see there's a 5x minus 4 in both groups. We're going to factor out or divide out that 5x minus 4. And so what we have is 5x minus 4 times 2x plus 1. Okay, just again really quick. So if you divide this by 5x minus 4, you're left with 2x.

If you divide this by 5x minus 4, you're left with 1. You can check your work. If you distribute the 5x minus 4 as a whole, as a group, to the 2x, you get back this term. If you distribute this all to 1, you get back this term. So you can check to see if you can factor any further, but it looks like that's as far as we can take this one.

Transcript for:Introduction to Factoring Polynomials

Transcript for:
Introduction to Factoring Polynomials