Video Transcript: Multiplying Binomials

Multiplying monomials 

Let's say that we wanted to multiply five x squared and, I'll do this in purple, three x to the fifth,  what would this equal? Pause this video and see if you can reason through that a little bit. All  right, now let's work through this together. And really, all we're going to do is use properties of  multiplication and use properties of exponents to essentially rewrite this expression. So we can  just view this, if we're just multiplying a bunch of things, it doesn't matter what order we  multiply them in. So you can just view this as five times x squared times three times x to the  fifth, or we could multiply our five and three first, so you could view this as five times three,  times three, times x squared, times x squared, times x to the fifth, times x to the fifth. And now  what is five times three? I think you know that, that is 15. Now what is x squared times x to the  fifth? Now some of you might recognize that exponent properties would come into play here. If I'm multiplying two things like this, so we have the some base and different exponents, that  this is going to be equal to x to the, and we add these two exponents, x to the two plus five  power, or x to the seventh power. If what I just did seems counterintuitive to you I'll just remind  you, what is x squared? x squared is x times x. And what is x to the fifth? That is x times x times  x times x times x. And if you multiply them all together what do you get? Well you got seven x's  and you multiply them all together and that is x to the seventh. And so there you have it, five x  squared times three x to the fifth is 15x to the seventh power. So the key is, is look at these  coefficients, look at these numbers, a five and a three, multiply those, and then for any variable you have, you have x here, so you have a common base, then you can add those exponents,  and what we just did is known as multiplying monomials, which sounds very fancy, but this is a  monomial, monomial, and in the future we'll do multiplying things like polynomials where we  have multiple of these things added together. But that's all it is, multiplying monomials. Let's  do one more example, and let's use a different variable this time, just to get some variety in  there. Let's say we wanna multiply the monomial three t to the seventh power, times another  monomial negative four t. Pause this video and see if you can work through that. All right, so  I'm gonna do this one a little bit faster. I am going to look at the three and the negative four and I'm gonna multiply those first, and I'm going to get a negative 12. And then if I were to want to  multiply the t to the seventh times t, once again they're both the variable t as our base, so  that's going to be t to the seventh times t to the first power, that's what t is, that's going to be t  to the seven plus one power, or t to the eighth. But there you go, we are done again, we just  multiplied another set of monomials. 

Multiplying monomials by polynomials: area model 

We're asked to express the area of the entire rectangle below as a trinomial. We have our  rectangle here and it's broken up into these three smaller rectangles. And we see for all of these rectangles, the height here is four units and then the widths are expressed in terms, or at least  the first two, are expressed in terms of x and then this last one has a width of two. So what's  the area of the entire rectangle? I encourage you to pause the video and think about it. What's  the area of this blue, this blue, it looks like a square, but let's just call it a rectangle, which all  squares are rectangles so that's safe. Well, it's going to be the height times the width. So the  area here is going to be the height, which is four, times the width, which is x squared. And then  to that, we want to add the area of this, I guess we could say this salmon colored rectangle and 

well that's going to be the height four times the width 3x. So we could say four times 3x, we  could write it like that, but what is 4 times 3x? Well, that's going to be 12x. You have 3x four  times, I have 12 xs, so that's going to be 12x. 12x is the area of this salmon colored rectangle.  And then, finally, the area of this green rectangle, we actually can figure out it exactly, we don't even have to express it in terms of a variable. Its height is four, its width is two, so the area's  going to be four times two, or eight. And we are done. 

Area model for multiplying polynomials with negative terms 

In previous videos, we've already looked at using area models to think about multiplying  expressions, like multiplying x plus seven times x plus three. In those videos, we saw that we  could think about it as finding the area of a rectangle, where we could break up the length of  the rectangle as part of the length has length x, and then the rest of it has length seven. So this  would be seven here, and then the total length of this side would be x plus seven. And then the  total length of this side would be x plus, and then you have three right over here. And what area models did is they helped us visualize why we multiply the different terms or how we multiply  the different terms. Because if we're looking for the entire area, the entire area is going to be x  plus seven, x plus seven times x plus three, times x plus three. And then of course, we can break that down into these sub-rectangles. This rectangle, and this is actually going to be a square,  would have an area of x squared. This one over here will have an area of seven x, seven times x.  This one over here will have an area of three x. And then this one over here will have an area of  three times seven, or 21. And so we can figure out that the ultimate product here is going to be  x squared plus seven x plus three x plus 21. That's going to be the area of the entire rectangle.  Of course, we could add the seven x to the three x to get to 10x. But some of you might be  wondering, well, this is all nice when I have plus seven and plus three. I can think about positive  lengths. I can think about positive areas. But what if it wasn't that way? What if we were dealing with negatives instead? For example, if we now try to do the same thing, we could say, all right, this top length right over here would be x minus seven. So let's just keep going with it, and let's  call this length negative seven up here. So it has a negative seven length, and we're not  necessarily used to thinking about lengths as negative. Let's just go with it. And then the height right over here, it would be x minus three. So we could write an x there for that part of the  height. And for this part of the height, we could put a negative three. So let's see, if we kept  going with what we did last time, the area here would be x squared. The area here would be  negative seven times x, so that would be negative seven x. This green area would be negative  three x. And then this orange area would be negative three times negative seven, which is  positive 21. And then we would say that the entire product is x squared minus seven x minus  three x plus 21. And we can, of course, add these two together to get negative 10x. But does  this make sense? Well, one way to think about it is that a negative area is an area that you  would take away from the total area. So if x happens to be a positive number here, then this  pink area would be negative, and so we would take it away from the whole. And that's exactly  what is happening in this expression. And it's worth mentioning that even before when this  wasn't a negative seven, when it was a positive seven and this was a positive seven x, it's  completely possible that x is negative, in which case you would've had a negative area anyway.  But to appreciate that this will all work out, even with negative numbers, I'll give an example, if 

x were equal to 10. That will help us make sense of things. So if x were equal to 10, we would  get an area model that looks like this. We're having 10 minus seven, so I'll put minus seven right  over here, times 10 minus three. Now, you can figure out in your heads what's that going to be.  10 minus seven is three. 10 minus three is seven. So this should all add up to positive 21. Let's  make sure that's actually occurring. So this blue area is going to be 10 times 10, which is 100.  This pink area now is 10 times negative seven. So it's negative 70, so we're gonna take it away  from the total area. This green area is negative three times 10, so that's negative 30. And then  negative three times negative seven, this orange area is positive 21. Does that all work out?  Let's see, if we take this positive area, 100 minus 70 minus 30 and then add 21, 100 minus 70 is  going to be 30, minus 30 again is zero, and then you just have 21 left over, which is exactly what you would expect. You could actually move this pink area over and subtract it from this blue  area. And then you could take this green area and then you could turn it vertical, and then that  would subtract out the rest of the blue area. And then all you would have left is this orange  area. So hopefully this helps you appreciate that area models for multiplying expressions also  works if the terms are negative. And also, reminder, when we just had x's here, they could've  been negative to begin with. 

Multiplying monomials by polynomials 

Multiply negative 4x squared by the whole expression 3x squared plus 25x minus 7. So if you  multiply anything times a whole expression, you really just use the distributive property to  multiply each term of the expression by the negative 4x squared. So we're going to have to  distribute this negative 4x squared over every term in the expression. So first, we could start  with negative 4x squared times 3x squared. So we can write that. We're going to have negative  4x squared times 3x squared. And to that, we're going to add negative 4x squared times 25x.  And to that, we're going to add negative 4x squared times negative 7. So let's just simplify this  a little bit. Now, we can obviously swap the order. We're just multiplying negative 4 times x  squared times 3 times x squared. And actually, I'll do out every step. Eventually, you can do  some of this in your head. This is the exact same thing as negative 4 times 3 times x squared  times x squared. And what is that equal to? Well, negative 4 times 3 is negative 12. And x  squared times x squared-- same base. We're taking the product. That's going to be x to the  fourth. So this right here is negative 12x to the fourth. Now let's think about this term over  here. This is the same thing as-- and of course, we have this plus out here. And then this part  right here is the exact same thing as 25 times negative 4 times x squared times x. So let's just  multiply the numbers out here. These were the coefficients. 25 times negative 4 is negative  100. So it'll plus negative 100, or we could just say it's minus 100. And then we have x squared  times x, or x squared times x to the first power. Same base-- we can add the exponents. 2 plus 1  is 3. So this is negative 100x to the third power. And then let's look at this last term over here.  We have negative 4x squared. So this is going to be plus-- that's this plus right over here. We  have negative 4. We can multiply that times negative 7. And then multiply that times x squared. I'm just changing the order in which we multiply it. So negative 4 times negative 7 is positive  28. And then I'm going to multiply that times the x squared. There's no simplification to do, no  like terms. These are different powers of x. So we are done.