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### Course: Operations and Algebraic Thinking 222-226 > Unit 2

Lesson 2: The distributive property & equivalent expressions- The distributive property with variables
- Factoring with the distributive property
- Distributive property with variables (negative numbers)
- Create equivalent expressions by factoring
- Equivalent expressions: negative numbers & distribution
- Equivalent expressions: negative numbers & distribution
- Equivalent expressions

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# Factoring with the distributive property

Sal shows how to factor the expression 4x+18 into the expression 2(2x+9). Created by Sal Khan.

## Want to join the conversation?

- how does this help me in real life?(87 votes)
- I'm wondering the same thing, along with quadratic equations, and Pythagorean theorem.(53 votes)

- Can someone tell me what I did wrong here with this equation? The equation is -2(-7k+4)+9=-13 I distributed -2 with -7k and 4 so when I got that I got 14k and -8 so then you put it back in the equation as 14k + -8 +9=-13 right? But then I got stuck with the -8 and 9 I can't figure it out and I have a test on it tomorrow. I need help! I need help with what to do from the step with the -8 and 9. If anyone can figure it out today that would be amazing!! I'm BEGGING YOU(20 votes)
- You just add the -8 and 9. So -8+9=1.

This gives 14k+1=-13. Subtract 1 from both sides to get 14k= -14.(18 votes)

- i dont get it everything doesnt make sense(8 votes)
- i see no real application of this strategy in making toast, taking a shower, or running outside. just sayin. not rlly necessary.(3 votes)
- If you continue to study math, I promise this will be useful! In real life, you might use this if you enter a field in physics, math, engineering, science, or computer science, but for now your main goal in learning this should be to be comfortable with it so you can pick up more difficult math concepts. Starting around Algebra 2 and Precalculus, factoring will become something that needs to come naturally in order to solve more difficult problems.(7 votes)

- For anyone who doesn't understand, its basically undoing the Distributive Property. Instead of Distributing and simplifying, its just figuring out the greatest common factor (GCF) and dividing GCF fro both the numbers.(6 votes)
- how would you do it with negative numbers in the problem(4 votes)
- Same process. Let's factor, say, -8x - 40. So, I can do this in two ways:

1. I can factor out an 8 from both terms. This gives 8(-1x) + 8 (-5). Taking the 8 common, we get 8(-x-5)

2. I can factor out -8 from both terms. This gives -8(x) + (-8)(5). Taking the -8 common, we get -8(x+5)

Both answers are correct, by the way. So, use whichever one you wish to(4 votes)

- Sometimes ours are negative, how can you tell that from a subtraction sign?(3 votes)
- negatives and minuses are the same operation, so it all depends on where it is in the expression or equation. Negatives generally come at the beginning of things (front of expression, front of parentheses, or front of denominator), so in the expression -2(-3)/-6 would all be considered negatives. If it is between things, it is considered as a minus, so 3 - 5 is minus. Then you have the special case of minus a negative such as 4 - (-4) which ends up as a positive 4+4.(5 votes)

- So it's just finding the greatest common factor and splitting the number?(3 votes)
- Hey JOSEPHM! It's fadethephaser2310!

So lets work on this equation: 10y + 8x

To simplify this equation we have to find a number that both of these numbers can be divided by (AKA finding a common factor). Both of these numbers can be divided by 2, so that will become the number on the outside of the parentheses.

Example:

2(insert number + insert number).

10y divided by 2 is 5y:

2(5y + insert number)

And 8 divided by 2 is 4x"

2(5y + 4x)

You can apply these steps to other equations as well. I hoped this helped and didn't confuse you more!

God Bless!(3 votes)

- Can anyone find me a person who uses this knowledge in their everyday life that isn't a math teacher or a mathematician? That would be very motivating...(4 votes)
- You will use it if you can look out for any problem involving an unknown (optimisation, finding out how much you need given a result, etc.). You can avoid it all you'd like, but this will make things much easier. I'm not saying you will use this in "everyday life", but it is essential to know :)(1 vote)

## Video transcript

What I want to do is start with
an expression like 4x plus 18 and see if we can rewrite
this as the product of two expressions. Essentially, we're going
to try to factor this. And the key here
is to figure out are there any common
factors to both 4x and 18? And we can factor that
common factor out. We're essentially
going to be reversing the distributive property. So for example, what
is the largest number that is-- or I could really say
the largest expression-- that is divisible into
both 4x and 18? Well, 4x is divisible
by 2, because we know that 4 is divisible by 2. And 18 is also
divisible by 2, so we can rewrite 4x as
being 2 times 2x. If you multiply that side,
it's obviously going to be 4x. And then, we can write 18 as
the same thing as 2 times 9. And now it might
be clear that when you apply the
distributive property, you'll usually end
up with a step that looks something like this. Now we're just going to
undistribute the two right over here. We're going to
factor the two out. Let me actually just draw that. So we're going to
factor the two out, and so this is going to
be 2 times 2x plus 9. And if you were to-- wanted
to multiply this out, it would be 2 times
2x plus 2 times 9. It would be exactly
this, which you would simplify as
this, right up here. So there we have it. We have written
this as the product of two expressions,
2 times 2x plus 9. Let's do this again. So let's say that I
have 12 plus-- let me think of something
interesting-- 32x. Actually since we-- just to get
a little bit of variety here, let's put a y here, 12 plus 32y. Well, what's the
largest number that's divisible into both 12 and 32? 2 is clearly divisible
into both, but so is 4. And let's see. It doesn't look like
anything larger than 4 is divisible into
both 12 and 32. The greatest common
factor of 12 and 32 is 4, and y is only divisible
into the second term, not into this first
term right over here. So it looks like 4 is the
greatest common factor. So we could rewrite each
of these as a product of 4 and something else. So for example, 12, we
can rewrite as 4 times 3. And 32, we can
rewrite-- since it's going to be plus-- 4 times. Well if you divide 32y by
4, it's going to be 8y. And now once again, we
can factor out the 4. So this is going to
be 4 times 3 plus 8y. And once you do more and
more examples of this, you're going to find
that you can just do this stuff all at once. You can say hey, what's
the largest number that's divisible into both of these? Well, it's 4, so let
me factor a 4 out. 12 divided by 4 is 3. 32y divided by 4 is 8y.