Factoring completely with a common factor

- [Instructor] So let's see if we can try to factor the following expression completely. So factor this completely, pause the video and have a go at that. All right, now let's work through this together. So the way that I like to think about it, I first try to see is there any common factor to all the terms, and I try to find the greatest of the common factor, possible common factors to all of the terms. So let's see, they're all divisible by two, so two would be a common factor, but let's see, they're also all divisible by four, four is divisible by four, eight is divisible by four, 12 is divisible by four, and that looks like the greatest common factor. They're not all divisible by x, so I can't throw an x in there. So what I wanna do is factor out a four. So I could re-write this as four times, now what would it be, four times what? Well if I factor a four out of four x squared, I'm just going to be left with an x squared. If I factor a four out of negative eight x, negative eight x divided by four is negative two, so I'm going to have negative two x. And if I factor a four out of negative 12, negative 12 divided by four is negative three. Now am I done factoring? Well it looks like I could factor this thing a little bit more. Can I think of two numbers that add up to negative two, and when I multiply it I get negative three, since when I multiply I get a negative value, one of the 'em is going to be positive and one of 'em is going to be negative. I can think about it this way. A plus B is equal to negative two, A times B needs to be equal to negative three. So let's see, A could be equal to negative three and B could be equal to one because negative three plus one is negative two, and negative three times one is negative three. So I could re-write all of this as four times x plus negative three, or I could just write that as x minus three, times x plus one, x plus one. And now I have actually factored this completely. Let's do another example. So let's say that we had the expression negative three x squared plus 21 x minus 30. Pause the video and see if you can factor this completely. All right now let's do this together. So what would be the greatest common factor? So let's see, they're all divisible by three, so you could factor out a three. Let's see what happens if you factor out a three. This is the same thing as three times, well negative three x squared divided by three is negative x squared, 21 x divided by three is seven x, so plus seven x, and then negative 30 divided by three is negative 10. You could do it this way, but having this negative out on the x squared term still makes it a little bit confusing on how you would factor this further. You can do it, but it still takes a little bit more of a mental load. So instead of just factoring out a three, let's factor out a negative three. So we could write it this way. If we factor out a negative three, what does that become? Well then if you factor out a negative three out of this term, you're just left with an x squared. If you factor out a negative three from this term, 21 divided by negative three is negative seven x. And if you factor out a negative three out of negative 30, you're left with a positive 10, positive 10. And now let's see if we can factor this thing a little bit more. Can I think of two numbers where if I were to add them I get to negative seven, and if I were to multiply them, I get to 10? And let's see, they'd have to have the same sign 'cause their product is positive. So let's see A could be equal to negative five, and then B is equal to negative two. So I can re-write this whole thing as equal to negative three times x plus negative five, which is the same thing as x minus five, times x plus negative two, which is the same thing as x minus two. And now we have factored completely.