Video Transcript: Special Products of Binomials

Special products of the form (x+a)(x-a) 

Let's see if we can figure out what x plus three times x minus three is, and I encourage you to  pause the video and see if you can work this out. Well, one way to tackle it is the way that we've always tackled it when we multiply binomials, is just apply the distributive property twice. So  first we can take this entire yellow x plus three and multiply times each of these two terms. So  first we can multiply it times this x. So that's going to be x times x plus three. And then we are  going to multiply it times, we can say, this negative three. So we could write minus three times, now that's going to be multiplied by x plus three again. And then we apply the distributive  property one more time. Where we take this magenta x and we distribute it across this x plus  three so x times x is x squared, x times three is three x, and then we do it on this side. Negative  three times x is negative three x and negative three times three is negative nine. And what does this simplify to? Well, we're gonna get x squared, and we have three x and minus three x so  these two characters cancel out, and we are just left with x squared minus nine. And you might  see a little pattern here, notice I added three and then I subtracted three and I got this, I got the x squared and then if you take three and multiply it by negative three, you are going to get a  negative nine. And notice, the middle terms canceled out. And one thing you might ask is, well,  will that always be the case, if we add a number and we subtract that same number like that?  And we could try it out. Let's talk in general terms. So if we, instead of doing x plus three times  x minus three, we could write this same thing as, instead of three, let's just say you have x plus  a times x minus a. And I encourage you to pause this video and work it all out, just assume a is  some number, like three or some other number, and apply the distributive property twice and  see what you get. Well, let's work through it. So first we can distribute this yellow x plus a onto  the x and the negative a. So x plus a times x, or we could say x times x plus a, that's going to be  x times x plus a, and then we're going to have minus a, or this negative a, times x plus a. So  minus, and then we're gonna have this minus a times x, plus a. Notice, all I did is I distributed  this yellow, I distributed this big chunk of this expression, I just distributed it onto the x and  onto this negative a. I'm multiplying it times the x and I'm multiplying it by the negative a. And  now we can apply the distributive property again. X times x is x squared, x times a is ax, and  then we get negative a times x is negative ax, and then negative a times a, is negative a  squared. And notice, regardless of my choice of a, I'm going to have ax and then minus ax. So  this is always going to cancel out. It didn't just work for the case when a was three. For any a, if I have a times x and then I subtract a times x, that's just going to cancel out. So this is just going  to cancel out, and what are we going to be left with? We are going to be left with x squared  minus a squared. And you can view this as a special case. When you have something, x plus  something, times x minus that same something, it's going to be x squared minus that  something squared. And this is a good one to know in general. And we could use it to quickly  figure out the products of other binomials that fit this pattern here. So if I were to say, quick,  what is x plus 10, times x minus 10? Well, you could say, all right this fits the pattern, it's x plus a  times x minus a, so it's going to be x squared minus a squared. If a is 10, a squared is going to be 100. So you can do it really quick once you recognize the pattern. 

Squaring binomials of the form (x+a) squared 

Let's see if we can figure out what x plus seven, let me write that a little bit neater, x plus seven 

squared is. And I encourage you to pause the video and work through it on your own. Alright,  now let's work through this together. So we just have to remember, we're squaring the entire  binomial. So this thing is going to be the same thing as: x plus seven times x plus seven. I'm  gonna write the second x plus seven in a different color, which is going to be helpful when we  actually multiply things out. When we see it like this, then we can multiply these out the way  we would multiply any binomials. And I'll first do it the, I guess you can say, the slower way, but  the more intuitive way, applying the distributive property twice. And then we'll think about  maybe some shortcuts or some patterns we might be able to recognize, especially when we are squaring binomials. So let's start with just applying the distributive property twice. So let's  distribute this yellow x plus seven over this magenta x plus seven. So we can multiply it by the  x, this magenta x, so it's going to be x, let me do it in that same color. So it's going to be  magenta x times x plus seven plus magenta seven times yellow x plus seven. X plus seven, and  now we can apply the distributive property again. We can take this magenta x and distribute it  over the x plus seven. So x times x is x-squared. X times seven is seven x. And then we can do it  again over here. This seven, let me do it in a different color, so this seven times that x is going  to be plus another seven x and then the seven times the seven is going to be 49. And we're in  the home stretch. We can then simplify it. This is going to be x-squared and then these two  middle terms we can add together. Seven x, let me do this in orange, seven x plus seven x is  going to be 14 x plus 14 x plus 49. Plus 49. And we're done. Now the key question is do we see  some patterns here? Do we see some patterns that we can generalize and that might help us  square binomials a little bit faster in the future? Well, when we first looked at just multiplying  binomials, we saw a pattern like x plus a times x plus b is going to be equal to x-squared, let me  write it this way, is going to be equal to x-squared plus a plus b x plus b-squared. And so, if both  a and b are the same thing, we can say that x plus a times x plus a is going to be equal to x squared, and this is the case when we have a coefficient of one on both of these x's, x squared's. Now in this case, a and b are both a. So it's going to be a plus a times x, or we can  just say plus two a x. Let me be clear what I just did. Instead of writing a plus b, I can just view  this as a plus a times x, and then plus a-squared, or that's the same thing as x-squared plus two  a x plus a-squared. This is a general way of expressing a squared binomial like this. A squared  binomial where the coefficients on both x's are one. We can see that's exactly what we saw over here. In this, in the example we did, seven is our a. So we got x-squared right over there let me  circle it. So we have this blue x-squared that corresponds to that over there. And then seven is  our a, so two a x , two times seven is 14 x. Notice we have the 14 x right over there. So this 14 x  corresponds to two a x, and then finally if a is seven, a-squared is 49. A-squared is 49. So in  general if you are squaring a binomial, you could, a fast way of doing it is to do this pattern  here, and we can do another example real fast, just to make sure that we've understood things.  If I were to tell you what is x minus, I'll throw a negative in here, x minus three squared, I  encourage you to pause the video and think about it. Think about expressing this using this  pattern. Well this is going to be, in this case our a, we have to be careful, our a is going to be  negative three, so that is our a right over there. So this is going to be equal to x-squared. Now  two a x, let me do it in the same colors actually, just for fun. So it's going to be x-squared. Now  what is two times a times x? A is negative three, so two times a is negative six. So it's going to  be negative six x. So, minus six x, that's two times a is the coefficient. And then we have our x 

there. And then plus a-squared. Well if a is negative three, what is negative three times  negative three? It's going to be positive nine. And just like that, when we looked at this pattern, we were able to very quickly figure out what this binomial squared actually is. And I encourage  you, you can do it again, with applying the distributive property twice to verify that this is  indeed the same thing as x minus three squared. 

Special products of the form (ax+b)(ax-b) 

Find the product 2x plus 8 times 2x minus 8. So we're multiplying two binomials. So you could  use FOIL, you could just straight up use the distributive property here. But the whole point of  this problem, I'm guessing, is to see whether you recognize a pattern here. This is of the form a  plus b times a minus b, where here a is 2x and b is 8. We have 2x plus 8 and then 2x minus 8. a  plus b, a minus b. What I want to do is I'm just going to multiply this out for us. And then just  see what happens. Whenever you have this pattern, what the product actually looks like. So if  you were to multiply this out, we can distribute the a plus b. We could distribute this whole  thing. Distribute the whole a plus b on the a and then distribute it on the b. And I could have  done this with this problem right here, and it would have taken us less time to just solve it. But I want to find out the general pattern here. So a plus b times a. So we have a times a plus b,  that's this times this. And then a plus b times negative b, that's negative b times a plus b. So I've done distributive property once, now I could do it again. I can distribute the a onto the a and  this b and it gives me a squared. a times a is a squared, plus a times b, which is ab. And now I  can do it with the negative b. Negative b times a is negative ab or negative ba, same thing. And  negative b times b is negative b squared. Now, what does this simplify to? Well, I have an ab,  and then I'm subtracting an ab. So these two guys cancel out and I am just left with a squared  minus b squared. So the general pattern, and this is a good one to just kind of know super fast,  is that a plus b times a minus b is always going to be a squared minus b squared. So we have an  a plus b times an a minus b. So this product is going to be a squared. So it's going to be 2x  squared minus b squared minus 8 squared. 2x squared, that's the same thing as 2 squared times x squared, or 4x squared. And from that, we're subtracting 8 squared. So it's going to be 4x  squared minus 64. 

Squaring binomials of the form (ax+b) squared 

We're asked to simply, or expand (7x + 10) ^ 2 Now the first thing I will show you is exactly what  you should NOT do, well there's this huge temptation. A lot of people will look at this and say  oh, that's (7x)^2 + 10^2. This is WRONG. And I'll write it in caps. This is WRONG! What your  brain is doing is thinking if I had 7x times 10 and I squared that, this would be (7x)^2 times 10^2.  We aren't multiplying here, we're adding 7x to 10. So you can't just square each of these terms. I just wanted to highlight, this is completely wrong, and to see why it's wrong, you have to  remind yourself that (7x + 10)^2 is the exact same thing as (7x + 10)(7x + 10). That's what it  means to square something. You're multiplying it by itself twice here. So this is what it is, so  we're really just multiplying a binomial, or two binomials, they just happen to be the same one,  and you could use F.O.I.L., you could use the distributive method, but this is actually a special  case: when you're squaring a binomial, so let's just think about it as a special case first then we  can apply whatever we learn to this. So we could've just done it straight here, but I want to  learn the general case so you can apply it to any problem that you might see. If I have (a+b) 

squared We already realised that it's not a squared plus b squared That is a plus b times a plus b. and now we can use the distributive property We can distribute this a + b times this a So we get, we get a times a plus b and we can distribute the a plus b times this b plus b times a plus b, and  we distribute this a we get a squared plus ab plus b times a is another ab And I'm just swapping  the order so it's the same as this. plus b times b which is b squared. These are the same or these are like terms. So we can add them. One of something plus another of that something will give  

you two of that something. 2 ab. We have a squared plus 2 ab plus b squared. So the pattern  here, the pattern here, if I have a plus b squared it's equal to a squared plus 2 times the product  of these numbers plus b squared. So over here I have seven x plus ten squared So this is going  to be equal to seven x squared seven x squared plus 2 times the product of seven x and 10. 2  times seven x times 10 plus 10 squared. So, the difference between the right answer and the  wrong answer is that you have this middle term here that you might have forgotten about if  you did it this way. And this comes out when you are multiplying all the different combinations  of the terms here. if we simplify this, if we simplify seven x squared That's seven squared times  x squared. So seven squared is 49 times x squared When you multiply this part out 2 times 7  times 10 which is 140 and then we have our x. No other x there. And then plus 10 squared. So  plus 100. And we are done.