Factoring completely with a common factor | Algebra 1 | Khan Academy

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. The way that I like to think about it is I first try to see if there is any common factor to all the terms, and I try to find the greatest common factor possible. Common factors to all the terms, so let's see, they're all divisible by 2, so 2 would be a common factor. But let's see, they're also all divisible by 4.

4 is divisible by 4, 8 is divisible by 4, 12 is divisible by 4, and that looks like the greatest common factor. They're not all divisible by x, so I can't throw an x in there. What I want to do is factor out a 4.

So I could rewrite this as four times… now what would it be? Four times what? Well, if I factor a four out of 4x squared, I'm just going to be left with an x squared. If I factor a four out of negative 8x, negative 8x divided by 4 is negative 2, so I'm going to have negative 2x. If I factor a 4 out of negative 12, negative 12 divided by 4 is negative 3.

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 2 and when I multiply, I get negative 3? Since when I multiply, I get a negative value, one of them is going to be positive and one of them is going to be negative. I could think about it this way: a plus b is equal to negative 2 and a times b needs to be equal to negative 3.

So let's see, a could be equal to negative 3 and b could be equal to 1 because negative 3 plus 1 is negative 2 and negative 3 times 1 is negative 3. So I could rewrite all of this as 4 times (x + negative 3), or I could just write that as (x - 3)(x + 1). And now I have actually factored this completely.

Let's do another example. So let's say that we had the expression negative 3x squared plus 21x 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 3, so you could factor out a 3. But let's see what happens if you factor out a 3. This is the same thing as 3 times… well, negative 3x squared divided by 3 is negative x squared, 21x divided by 3 is 7x, so plus 7x, and then negative 30 divided by 3 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 3, let's factor out a negative 3.

So we could write it this way: if we factor out a negative 3, what does that become? Well then, if you factor out a negative 3 out of this term, you're just left with an x squared. If you factor out a negative 3 from this term, 21 divided by negative 3 is negative 7x, and if you factor out a negative 3 out of negative 30, you're left with a 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 7, and if I were to multiply them, I get 10? And let's see, they'd have to have the same sign because their product is positive.

So, see, a could be equal to negative 5 and then b is equal to negative 2. So I can rewrite this whole thing as equal to negative 3 times (x + negative 5), which is the same thing as (x - 5)(x + negative 2), which is the same thing as (x - 2). And now we have factored completely.