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Multiplying binomials

Sal expresses the product (3x+2)(5x-7) as 15x²-11x-14. Created by Sal Khan and Monterey Institute for Technology and Education.

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Video transcript

Multiply (3x+2) by (5x-7). So we are multiplying two binomials. I am actually going to show you two really equivalent ways of doing this. One that you might hear in a classroom and it is kind of a more mechanical memorizing way of doing it which might be faster but you really don't know what you are doing and then there is the one where you are essentially just applying something what you already know and kind of a logical way. So I will first do the memorizing way that you might be exposed to and they'll use something called FOIL. So let me write this down here. So you can immediately see that whenever someone gives you a new mnemonic to memorize, that you are doing something pretty mechanical. So FOIL literally stands for First Outside, let me write it this way.....F O I L where the F in FOIL stands for First, the O in FOIL stand for Outside, the I stands for Inside and then the L stands for Last. The reason why I don't like these things is that when you are 35 years old, you are not going to remember what FOIL stood for and then you are not going to remember how to multiply this binomial. But lets just apply FOIL. So First says just multiply the first terms in each of these binomials. So just multiply the 3x times the 5x. So (3x. 5x). The Outside part tells us to multiply the outside terms. So in this case, you have 3x on the outside and you have -7 on the outside. So that is +3x(-7). The inside, well the inside terms here are 2 and 5x. So, (+2.5x) and then finally you have the last terms. You have the 2 and the -7. So the last terms are 2 times -7. 2(-7). So what you are essentially doing is just making sure that you are multipying each term by every other term here. What we are essentially doing is multiplying, doing the distributive property twice. We are multiplying the 3x times (5x-7). So 3x times (5x-7) is (3x . 5x) plus (3x - 7). And we are multiplying the 2 times (5x-7) to give us these terms. But anyway, lets just multiply these out just to get our answer. 3x times 5x is same thing as (3 times 5) ( x times x) which is the same thing as 15x square. You can just do this x to the first time to x to the first. You multiply the x to get x squared. 3 times 5 is 15. This term right here 3 times -7 is -21 and then you have your x right over here. And then you have this term which is 2 times 5 which is 10 times x. So +10x. And then finally you have this term here in blue. 2 times -7 is -14. And we aren't done yet, we can simplify this a little bit. We have two like terms here. We have this...let me find a new color. We have 2 terms with a x to the first power or just an x term right over here. So we have -21 of something and you add 10 or in another way, you have 10 of something and you subtract 21 of them, you are going to have -11 of that something. We put the other terms here, you have 15... 15x squared and then you have your -14 and we are done. Now I said I would show you another way to do it. I want to show you why the distributive property can get us here without having to memorize FOIL. So the distributive property tells us that if we 're... look if we are multipying something times an expression, you just have to multiply times every term in the expression. So we can distribute, we can distribute the 5x onto the 3..., or actually we could...well, let me view it this way... we could distribute the 5x-7, this whole thing onto the 3x+2. Let me just change the order since we are used to distributing something from the left. So this is the same thing as (5x-7)(3x+2). I just swapped the two expressions. And we can distribute this whole thing times each of these terms. Now what happens if I take (5x-7) times 3x? Well, thats just going to be 3x times (5x-7). So I have just distributed the 5x-7 times 3x and to that I am going to add 2 times 5x-7. I have just distributed the 5x-7 onto the 2. Now, you can do the distributive property again. We can distribute the 3x onto the 5x. We can distribute the 3x onto the 5x. And we can distribute the 3x onto the -7. We can distribute the 2 onto the 5x, over here and we can distribute the 2 on that -7. Now if we do it like this what do we get ? 3x times 5x, that's this right over here. If we do 3x times -7, that's this term right over here. If you do 2 times 5x, that's this term right over here. If you do 2 times -7, that is this term right over here. So we got the exact same result that we got with FOIL. Now, FOIL can be faster if you just wanted to do it and kind of skip to this step. I think its important that you know that this is how it actually works. Just in case you do forget this when you are 35 or 45 years old and you are faced with multiplying binomial, you just have to remember the distributive property.