Hey, it's Kaeleigh. In this lesson, we're going to learn how to factor. Make sure you
already are an expert at using the distributive property before we start this lesson. Let's go!
Factors are the numbers we multiply together to get another number. For example, if we have the
equation 4 times 3 equals 12, 4 and 3 are the factors of 12. We can also find the factors of
expressions like 6y. The factors would be 6 and y, since when we multiply them together, we get 6y.
Now, what if we have a more complex expression like 2x plus 6?
How can we factor this? Well, one way is to factor each term separately first. So what
are the factors of 2x? Remember, that means what do we multiply together to get 2x? Yeah, a 2 and
an x. Now what about for 6? What can we multiply together to give us 6? What are the factors? Well,
we could multiply 1 and 6 together to get 6, or we could multiply 2 and 3 together to get 6.
Now we want to take a look at our factors here, the ones in blue for 2x and the ones in orange
for 6, and see if there's any matching factors. Do you see any matching factors?
I do! I see a 2 right here and a 2 right here. So, since the factors for 6, 2 and 3,
give us a matching factor with 2x, we aren't going to worry about the 1 and the 6 this time.
So now let's take a look, 2 and x and 2 and 3 for this expression.
That means this is really saying 2 times x plus 2 times 3. So, from our expression like this,
what we're gonna do is we're gonna factor a 2 out. So we're taking these
2's here out over here, and then we're going to have parentheses for what's left over.
Now, if we took this 2 out, what do we have left? Just this x. And then I have a plus sign,
and I took this 2 out, and what do I have left? Just this 3 here. So this is telling
me that my factored form of 2x plus 6 is equal to 2 times x plus 3. So my factors of 2x plus 6 are
2 and x plus 3. And when I multiply them together, I should get 2x plus 6.
But I can test that, right? If I want to know if 2x plus 3 is equal to 2x plus 6, what can I do?
I can use the distributive property. So I'd multiply 2 times x, and that will give me 2x.
And then I'm going to multiply 2 times 3, and that will give me
6. And check it out, same expression. So that means we factored it correctly.
This shows us that factoring is the opposite of expanding or using the distributive property.
Let's do a few more examples. Here I have two expressions to factor. And let's start
with 4m minus 2. So, let's factor my terms separately. 4m, how can I factor that? Well,
4 times m would give me 4m. And how about my 2? How do I factor out my 2? A 1 and a 2.
Now, do you see any matching factors here?
No, me neither. But I can factor this 4 down. What times what will give me 4? Well,
1 times 4 obviously, but this 4 isn't working. So, what else do I got? 2 times 2 will get me 4.
Now do you see any matching factors? Yes! We have a 2 here and we have a 2 here.
So let's rewrite our expression using its factors. We would have 4m is the same as 2 times 2
times m minus 1 times 2. And now what I'm gonna do is take the 2's out of my expression. So I'm
factoring it out. That means I'm putting my 2 here and I'm getting out my parentheses.
So now, in my first term, I took this 2 out here, but I'm still left with two factors, a 2 and an
m. And then minus my second term. I took out this 2, it's right here, but I'm still left with my 1.
So this is my factored form of the expression 4m minus 2. It's 2 all times 2m minus 1.
I can double-check and make sure I factored this right using the distributive property. So I'd have
2 times 2m, which would give me 4m, minus 2 times 1, which would give me 2. I double-check. That
expression matches that expression, so that's it. And that's the correct factored form. Nice!
Now let's go over to our second one here. We have 6g plus 12. So I'm going to start by factoring 6
and g. I get a 6 times g. And how about for my 12? There's a few ways I could factor 12: 1 and 12,
2 and 6, 3 and 4. Do we see any matching factors here? Yes! We have a 6 here and a 6 here.
So, for my 12, I have a few different sets of factors, but since the 6 is the only matching one,
I'm going to forget about this one and forget about this one for now.
So let's rewrite these using their factors. So for my first term, I'd have 6 times g plus 2 times 6.
So now I'm going to factor out my matching factors. So the 6's are my matching factor.
I'm taking the 6's out, and I'm putting my parentheses down. So my first term,
I factored this 6 out, it's right here, but I'm still left with my g. Plus, my second term here,
I factored that 6 out. It's here outside my parentheses, but inside I still have a 2.
So the factored form of 6g plus 12 is 6, all times g, plus 2. I can check this using what property?
The distributive property! So 6 times g gives me 6g, plus 6 times 2 gives me 12. Double-check that
expression matches that expression. Boom! That's the correct factored form. Nice!
All this factoring might make you wonder, why do we actually need to know how to do this? There are
a ton of reasons learning to factor is important, but this lesson is just the very beginning.
Eventually, you will solve complex equations like 4x squared plus 2x equals negative 2,
and even plot graphs or projectiles like knowing when a piano launched
from a rocket will crash into the ground. That's why knowing how to factor is so cool.
Now you can begin factoring and using the distributive property
on your own. Have fun, and remember to always be clever!