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# Notes for Lesson 3: Polynomials

## Grant is Glum

He had a huge wait list of people who had, in total, asked for 87 wiper blade sets and a turn for the worse. One of his customers called, saying that she had just gotten contacts and she would no longer be needing the wipers or the nozzles that she had requested. Another person emailed him to say that he had gotten laser eye surgery and he wanted to cancel his order as well. Grant was beside himself. Not only was he sad that he had lost some customers, he was frustrated that he would never be able to target the population that didn't wear glasses.

## Quantifying Losses

Grant also didn't know how these ordered cancellations would affect the total amount of money he would make. We can't really help Grant with his business issues, but we can help him with the math he needs to know about. So which of these expressions down here should replace the question mark right here, to create an equation for the future earnings that Grant will have after all of these orders are paid for? Remember that we need to take into account, both the money that he thought he was going to make from all of the orders he originally received, and then also, the money that he will no longer make from the orders that have now been cancelled. Please assume that money from all original orders, and money from cancelled orders down here are positive when written on their own.

## Quantifying Losses

So we know that the total money Grant will earn from his sales is going to have to account for both the money he thought he was going to make from all of the wipers and nozzles he thought he was going to sell and also for the money that he will no longer make from the canceled orders. So that means that we can already rule out two of our answers. These two right here, the second and the forth, each of which only takes into account one of those two properties we need to have involved in our equation. Now we also know that Grant is not going to make as much money as he originally thought that he would because we need to remove the money that he would have gotten from the cancelled orders. If you want to remove a quantity you want to subtract it, not add it. So that means that this first answer choice, is the correct one. If we picked the third answer, then Grant would end up with more money, since we said that both of these quantities are positive after the orders were cancelled than he would have before any orders were cancelled. So that doesn't make sense. I know that the wording of things may have been a tiny bit confusing here, but in the end this is all going to help Grant out a ton, so it's worth our effort.

## Canceled Orders

Selling 3 wiper sets would have given Grant 3 times the cost of 1 wiper set, which would be 3w dollars. And selling 12 nozzles would have given him, well, 12n dollars. We need to add these 2 quantities together to get the total amount of money that Grant would of made from all these canceled orders. So our answer is just 3w + 12n.

## Total Future Earnings

Remember that the equation we had before. If the amount of money that Grant would earn if all of his original orders had gone through, was equal to 87w plus 72n. But this was before any orders had been canceled. And in the last quiz, we just found out how much money Grant would have earned from the orders that are now canceled. And that was just equal to 3w plus 12n dollars. A couple of quizzes ago, we also found an equation for Grant's total future earnings, taking his canceled orders into account. Of course, it was a pretty general equation, that didn't have any variables numbers in it yet. They said that his total future earnings will be equal to the money from all the original orders, minus the money he would have made from the orders that are now canceled. So let's combine all this information, using substitutions, into a single equation, for Grant's total future earnings. Earnings. So which of these expressions down here would correctly replace the question mark to come up with an equation for Grant's total future earnings, taking the cost of each wiper and nozzle into account.

## Total Future Earnings

This is just another case we need to substitute in new values for these two quantities, money from all original orders and money from canceled orders into this expression using these other two equations. As we learned about substitution very early on in the course, if we have a system of equations that we know is allowed to work together and one quantity in one of the equations is equal to another quantity in another one of the equations. We can replace that quantity with the other way to express it. So in the case of these equations that means we can replace money from all our original orders right here with this expression, Since this equation tells us that, that is equal to the money from all the original orders. We can do the same thing with the money from the canceled orders or the amount of money that Grant would have gotten if these orders hadn't been canceled. So we'll replace money from canceled orders with 3w+12n. So this is what the equation looks like if we just replace that one word money from canceled orders with the mathematical expression we know that it's equal to. But we need to do one thing to change this equation. We need to add parenthesis around 3w+12n. Now the reason we need to do that is because we know what we want to subtract the entire amount of money that Grant would have earned from the orders that are now cancelled. So to show that this is one quantity, one sum of money, the money he would have earned from the wipers and the money he would have earned from the nozzles, we put parenthesis around it.

## Combine Like Terms

As it is, this equation could be useful for Grant, but I think that it could be more helpful if it looked a little bit cleaner. So, let's simplify. But wait, how do we deal with this minus sign and these parentheses around this part of the expression? We see like terms here so we know that we will be able to simplify this somewhat, and even though, we haven't talked about exactly how to do this, just give it a try. If it doesn't work out, no problem. We'll talk about it in a second. So type your answer for the simplified version of this expression into this box and please remember to simplify as much as you can. Good luck.

## Combine Like Terms

I said earlier that the reason we needed to put parenthesis around 3w + 12n, or the total amount of money that Grant would have gained from the orders that have now been canceled, was because this is one sum of money, we want to subtract the whole thing, not just part of it. If we didn't have parenthesis here then we would only be subtracting the money that he would have earned from the wipers that are now canceled, but we would still be adding the amount of money that he would have gotte from the nozzle orders that are now canceled. But we know that both of these sets of orders have been cancelled, so we need to make sure that both of them are subtracted from the original sum of money. So let's put those parentheses back. However, I don't really want these parentheses here in the end version of this expression, so we need to get rid of them somehow. Intuitively like I just said, you want to subtract both this quantity 3w the money from the cancelled wiper orders and this quantity 12n the money from the cancelled nozzle sales. So we want to subtract both of them and if I get rid of these parentheses I should just be able to switch this plus sign to a minus sign and have the expression that. That I want. This makes sense. Here's our original amount of money that Grant would've earned from all of the orders that he had. And then we take away both the wipers that he's not selling anymore and the nozzles he's not selling anymore. That makes sense. Now the problem is simple, you've done this so many times before. You just need to rearrange the order of the terms, so that like terms are now. Next to each other and then simplify. So, first we rearranged the order of the terms so that w terms are next to each other and the n terms are next to each other and then we just add their coefficients together and multiply it by the proper variable. So 87-3 is 84 and the variable there is w and 72-12 is 60 and the variable there is n. So here's our final answer. Answer, 84w + 60n. Awesome job. That quiz had a bunch of steps in i t, and some new material. So if you got it even partially right, that's really great.

## Numerical Example

To talk more about subtracting polynomials, let's start off with a pretty simple example using just numbers. So, let's say that we had the numerical expression, 10-(2+3). What is this equal to? Just put a number in this box.

## Numerical Example

So the answer to the problem, as you may have found out is 5. But there are actually two different ways that you can solve this problem. Let's go through both of them. So, if you remember our order of operations acronym, PEMDADS> This tells us that, before we do any subtraction, which we'll have to do right here. We need to deal with whatever's happening inside parentheses. So if we were following this set of rules, we would say that this is equal to 10 minus dealing with what's inside the parentheses, 2+3 is 5, and then we would do the subtraction, 10-5 is 5. However, as we saw in dealing with the equation that we gave to Grant, there's a way to get rid of parentheses, so that we can solve this problem in different way. Let's pretend that the numbers that we have up here actually refer to some objects. Let's say these circles that I have done here right now. So, here are ten circles and I want to figure how to modify these super circles in accordance with this expression. So we have ten vice, and then you want to get rid of 2+3 of them. So if we did the problem the original way, where we have 10-5 circles. Then we would just count out five of these 1, 2, 3, 4, of them. And then in the end we're left with 1, 2, 3, 4, 5, five circles. However, let's talk about how to do the problem in the other way. Instead of grouping the circles that we want to get rid of as one whole group, as we did in the first way, so that's by dealing with what's inside the parenthesis first, and circling all five of the circles we want to get rid of, let's think about these as two separate groups. So, we know that we want to get rid of two circles. So that's just these two and we also want to get rid of three circles. Now instead of merging these into one group like we did the first way, why don't we just get rid of each of them separately since we know that we need to get rid of this whole thing. So first we'll get rid of these two and then we'll get rid of these three as well and in the end we' re left with the same five circles that were when we did the problem the first way. So what we really did was subtract two circles. So 10 minus 2 circles and then we subtracted the 3. So this gave us, our final five circles.

## Subtracting Polynomials

Since we are doing algebra in this course after all let's look at an expression that is similar to the last one we looked at but actually has a variable it in. So here we have 4 - (x+9). What do you think this is equal to? Please make sure to simplify your answer as much as you can so I don't want to see any parenthesis in here and I also want to see all like terms combined.

## Subtracting Polynomials

So the way that we deal with this problem is actually exactly like we did when we were doing 10-(2+3), but this time we can't change what's inside the parenthesis. There's no way that we know of to simplify x+9, since these are not like terms. So that means, that we can't go about doing things with the first method of PEMDAS, or at least the p here doesn't affect the way that we can do the problem, since we can't change what's inside the parenthesis. So, that means we're stuck with this second method. Remember down here that meant applying the negative sign to both the 2 and the 3 so we ended up subtracting each of them from the 10. So let's try to do the same thing up here. We want to subtract both the X and also the 9 so we'll have 4 minus X and then also minus 9. Rearrange and then combine like terms 4-9 is -5-x so we have our final answer of -5-x.

## Distributive Property

So, it's pretty straight forward to think about applying the negative sign to both the x and the 9 when we're talking abut subtracting expressions from one another, but what we're really doing here is using an example of a much broader property. And that property is the Distributive Property. So, we're distributing this negative sign to both the x and to the 9 when we say that we want to subtract them both from the 4. So, I'm going to show you a slightly different way of looking at how to distribute that negative sign and what distributing a negative sign really means. First things first, we know that when we subtract, like we're doing here, that's actually the same as adding the negative of the thing that you want to subtract. So, for example, if we have 5-3, that's actually equal to 5 plus -3. Now, that may seem like we're working backwards but hang in there, this is going to be helpful. So, let's do the same thing here. This is equal to 4 plus the negative of this entire quantity. However, when we're taking the negative of something, what we're really doing is multiplying it by -1. So, in order to get -3, we actually have to say -1 times +3. So, that means that if we want -x+9, we actually need -1 times x+9. You'll notice that I didn't write a multiplication sign between this coefficient, -1, and the parentheses that's around x+9, because remember, there's an invisible multiplication sign implied between those when there's no space between them. So, I'll just get rid of that. So here, it's a bit easier to see explicitly why the Distributive Property is being applied in this case. We have a -1 that we're multiplying each term inside the parenthesis by. So, both x and -9 are each being multiplied by -1. The Distributive Property basically says that if you have some number or variable, let's call it a, and you want to multiply that by some number of terms that are inside parentheses, let's just pick two to be simple and call them b and c. And we'll just add them, although of course, c can be negative, and you could be subtracting or b could be negative. If you want to get rid of the parentheses, you'll end up with a being multiplied by each of these terms. So, the a multiplies the b and that's added to a times c as well. So, you can see why we use the word distribute here. The multiplication of the a affects the b and also affects the c. So, you have a times b, original operator that was between b and c, the plus sign, a times c. So, if we use that property in this case, we leave our 4+ and then we have -1 times x. Write the sign that's between the two terms inside the parentheses, -1 times 9. However, we know that -1 times x is just -x and -1 times 9 is also just -9, so we end up with when we are quote-unquote distributing this minus sign to both terms in here and then, of course, we could rearrange and simplify again.

Now, since it's my job to make your life difficult, I'm going to give you a nice complicated expression here. You have (7x^2+2x+1), all inside parentheses, and I'd like you to add that to this other quantity inside parentheses, (4x^2-2x+4). And remember, I'd also like you to simplify as much as you can, so no more parentheses and all like terms combined. I know this looks complicated, but you know how to do this, and if you don't get it right, no big deal.

Okay, so we have 2 polynomials that we are trying to add together. And this may sound silly, but if we're adding one quantity to another, we're actually adding +1 times this quantity. Anything, let's say some number a, times 1 is just equal to itself. So a times 1 is a, and 5 times 1 is 5, and 4x^2 times 1 is 4x^2, and -2x times 1 is-2x. So having the parentheses here with the plus sign in front of them doesn't change the value or the sign of any of the terms inside the second set of parenthesis. So we can just write it but without the parenthesis. There's also nothing happening to the first set of parenthesis, at least not yet, so we can just rewrite it without the parenthesis as well. And now, we have a thing that we have a ton of practice we're dealing with. We just need to rearrange our terms and simplify. Once we've rearranged we combine our now adjacent like terms. So for x^2 term we have 11x^2. And now I'm dealing with the x to the first terms, +2 -2 is just zero, so these cancel each other out. And we don't have any term in the final answer with an x^1 in it. Then we just have 1+4 so our final answer is 11x^2+5. I just love doing these problems so much. because you start out with something that's so complicated and messy looking, and you end up with a really, really beautiful simple answer.

## Another Subtraction

Let's subtract these two polynomials. So what is your final answer if I ask you to subtract 4x^2-2x+4 from the quantity Remember to simplify.

## Another Subtraction

So, as we talked about in our example before, we need to use the distributive property here to figure out how this minus sign affects all of the terms inside the second set of parentheses. First off though, since nothing is happening outside of the parentheses, on the front end at least, for this first part of our expression, we're just going to rewrite that without the parentheses. And then, we remember the trick that we talked about before that a minus sign actually means plus -1 times whatever you want to subtract. So instead of the minus sign right here, that's what I am going to write, we're adding -1 times this entire thing in parentheses. And now, we need to use the distributive property to multiply -1 by every term in here, this is going to affect the 4x^2 and the -2x and also the 4. So, going one step further, you'll notice that for right now, I've let there be an addition sign between all of the terms over here, and then their sign is being determined within the term before I change any of these to subtraction signs if they're going to be them eventually. So, now we can carry out the simplification within each term. Remember that if you have two negative numbers multiplied together, they equal a positive number. So, -1 times -2 is actually equal to positive 2. And now, since we have two terms where I'm adding a negative, I'm going to switch these to negative signs or minus signs. And then, last but not least, we rearrange and combine like terms. So our final answer is Awesome job. There are a ton of steps as you can see in this problem, and there are a lot of places where it's really easy to make a silly mistake, so that's why it's super important that we write every single step down. I know this probably feels really tedious, but, in the end, as you keep practicing getting more proficient, it's going to be the thing that gives you the right answers. So please just keep taking the time to write out all of your steps and don't take any shortcuts quite yet.

## Distributive Property Review

So we already talked about the general version of the distributive property is that if you have some variable, or number, or expression, or really anything in the slot where a is and you want to multiply that by some set of terms that are added together Inside parentheses. They could also, of course, be subtracted. Then the factor that you want to multiply everything by distributes its multiplication to each term that's inside the parentheses. So we end up with a b, or ab over here, + a c, ac over here. Now, we already saw how we use the distributive property when we want to subtract polynomials from one another. But in those situations, we let a = -1, and clearly, a could equal many other things besides -1. So, let's try something out, where a is a different number.

## Warming up

So, to ease us in, to working with the distributive property, in many different ways, let's just start with a simple number example. What do you get if you multiply 3 by, the quantity 2+10. Just write a number answer in this box.

## Warming up

So once again, there are actually two ways to solve this problem. The first way, using PEMDAS again, is to add what's inside the parentheses 2+10, and then do the multiplication. It's one to split this into our two different methods. We can either say that we end up with 3 times the quantity 2+10 or 12. And then 312 is just equals to 36. Or as we have learned before we can use the distributive property. So, if we want to do that, we first multiply 3 by 2 and then add whatever we get for that, 2 3 times 10. So if we write that out if 32+310 and then we evaluate each of these terms individually, 32 is 6 and the same answer.

## Visualizing Distribution

I'd like to take a minute to give you a visual of why the distributive property works the way that it works. So, let's go back to drawing our beautiful circles. So let's say we have a group of 2 circles and also a group of 10 circles. And, according to this equation, what we're going to want to do is both add them together and then triple them. So if we were to solve this problem the first way that we talked about, we would do this addition inside the parentheses first, and then our multiplication. So you would take these two groups, the 2 and the 10 circles, and merge them into one group. Great. So now we have 12 circles in one big group. And then, the next thing I want to do is multiply this group by 3, so that means we want 3 sets of this group of circles. So right now, we have one set and I can add 1 more, which make it 2 sets and then I can add a third one. Remember, multiplication is just repeated addition. So now, I have 3 groups of these 12 circles together and that's it. I can count these up and if I decide, I'd do that right now, which I won't, because I'm lazy, we would see that we have 36 circles. Okay, great. So, that's the first method for solving this problem. However, we also have that 2nd method of solving problems like this and that's using the distributive property. So if we want to use the distributive property, then the first step is going to be to multiply each of the terms that's inside the parentheses by the 3. So let's write down method 2. First, we're going to say 3 times 2, then we're going to add that to 3 times 10, just like we saw in the last quiz. Starting with the group of two circles, I wanted to triple this like we see in this 1st term. And then, as we see in the 2nd term, I want to triple this group as well. Great. So now I have two groups, one group of 6 circles which is 3 times 2, and one group of 30 circles which is 3 times 10. And then, I want to add those two groups together. So we need to merge this set of circles with this set of circles, and we end up with the exact same picture that we got using the first method. Once again, we have 36 circles all in one group. I hope this has given you some insight into why we're allowed to distribute multiplication to terms that are added together inside a parentheses.

## Time To Distribute

So to use your new found knowledge of the distributive property let's have a quiz. What is 3 times all this stuff in parentheses? x^2+5x-7. Try not to look in this upper right hand corner of the screen if you don't need to. The distributive property is still written out there, but see if you can do it without.

## Time To Distribute

So this problem may or may not have thrown you off a little bit, because as we can count, there are 3 terms inside these parentheses and in the problems we've done before. And in the rule that I wrote up here you only saw 2 terms added together. However it does not matter how many terms are inside a parentheses a multiplication is still going to distribute to every single one of them. So in this case the 3 is going to multiply the x^2 and also the out. So distributing 3 to each of these terms, we get 3x^2+3(5x)+3(-7), and then we can simply within each term. And we get a final answer of 3x^2+15x-21.

## Variable Distribution

So let's continue with our practice of the Distributive Property. Can you tell me what -5y(y+z) is equal to? Remember to simplify within each term as much as you can. And also remember if you need to write an exponent, let's say you need to write 2^3, you would type that in as 2^3. So, write the carrot sign before any exponent that you need to type.

## Variable Distribution

So once again we're just going to use the distributive property in the same way as we've done it many, many times before. The difference this time and what may have confused some people, is that our factor, our little a up here. If you can see, is actually two things multiplied together. However that does not change the way that we use the distributive property at all. As always the -5y, the factor out in front, is going to multiply each term inside here, both the y And the z. So our first term is going to be -5y y, and our second term is just going to be -5y z. Now again, I've just put an addition sign between the two terms for simplicity right now, so that I can determine the sign of each term within the term itself, and then change this to a - sign if I need to. That's just a lot easier for me personally to not get my signs mixed up. So now we simplify within each term. And we end up with -5y^2 - 5yz. So you can see that no matter how many things are multiplied together to create the factor that we have out in front of our parentheses, they all multiply each of the terms within the parentheses.

## Challenge Problem

So now that you've had a fair amount of practice with the distributive property, I'm going to give you a challenge problem. In fact, I'm going to officially declare that this is a challenge problem by writing challenge and I'll give you a few exclamation points just to reinforce my point that I think this is really hard. Here's a pretty complicated expression if I do say so myself. We have x+4-3x times the entire quantity, y+5-2x+7x^2, and then subtract it from all of that, 6y. So, this is gross and messy but you have all the tools that you need to simplify this and make it look relatively pretty. So take your time, write everything out on paper, and seriously, make sure that you don't skip any steps. Or, at least I know, that when I did this problem, if I hadn't written anything down, I would have made a bunch of mistakes. So, take your time. Good luck.

## Challenge Problem

Judging from how complicated this looks right now, they're probably going to be a lot of steps involved in simplifying this expression. So, let's just get started. The first thing I'm going to do is use the Distributive Property to distribute multiplication of -3x to every term inside the parentheses. Remember that it doesn't matter how many terms are inside the parentheses, we still multiply every single on of them. So, for now, that's the only thing that I'm going to change. I'm going to leave everything else, all the other terms in the expression, exactly as they are. So now, we have this long expanded expression. I'm really sorry. I had to erase your challenge warning sign, but know that I still think this is a challenge problem. Now, we'll simplify it within each term. You'll notice that I haven't switched any of these operator signs to minus signs yet so I'll do that in this next step. So, wherever I'm adding a negative term, I'm actually just subtracting that term. So, instead of plus -3xy, I'll actually just have -3xy. And now finally, we get to rearrange our terms so that like terms are next to each other and simplify. And so, here's our final answer. You still have six terms, which just goes to show you sometimes, you can't do anything more to make an expression look any prettier. But it definitely is a bit less complicated than it was at the beginning. I know this was a really tough problem and again, there were many, many opportunities for little mistakes to be made. So, if you got even part of this right, especially practicing the distributive property, that is awesome. So, if you made a little mistake, go back and try it again. The Distributive Property will keep popping up at different places throughout the course so it's super important that you really, really understand it. Although, probably not all the problems we do will be quite as complicated as this one.

## Cheering up Grant

So the last equation that we'd created for Grant to tell him how much he would earn in the future, once people actually paid for the glasses, wipers, and nozzles they had ordered, was that his total future earnings = 84w, where w is the price per wiper set plus 60n, where n is the price per nozzle. However, there's been another change in the situation. In considering the last change that happened to Grant, which was the cancellation of orders that made him really, really sad, this is a super positive update. First, Grant got a telephone call from an elderly wealthy gentleman, named Mr. Belvedere, who happens to be super interested in glasses wipers and in the nozzles that Grant has developed to go along with his wiper blades. He was so enthusiastic about Grant's inventions that he offered to triple all of the orders that Grant had brought in so far, which means that Grant's total future earnings, which we know before were 84w + 60n, will be multiplied by 3. Then a second piece of good news came. A school principal called Grant and let him know that she wanted to order, one wiper blade set and one pair of nozzles, for each of her 150 students who wear glasses. This however happened after Mr. Belvedere decided to triple the orders that had come in so far. So his tripling does not apply to this new batch orders from the school.

## Describing the Good News

What we want to do now, is take into account, these two updates to Grant's total future earnings. We know that the first thing that happens, is that Mr. Belvedere will triple the orders that Grant had initially. And then after that tripling happens, a school orders one wiper set and two nozzles for each of its happens before step two. So taking those two things into account what is a new equation for Grant's future earnings? So, in other words, how do we modify the equation that we had before for his total future earnings, 84w+60n, to reflect these two changes to the number of orders that he has, and therefore the amount of money that he's going to make? Please choose the best choice from these four.

## Describing the Good News

To find the updated total future earnings, we're going to start by writing down the initial total future earnings we had. So, this is our baseline, this is what we start out with, and now, I'm going to show how these two changes affect the equation. The first thing that happens is that Mr. Belvedere triples the amount of money that Grant will eventually make off of these first orders. So, that means we need to multiply this quantity, 84w+60n, by 3. So I'm going to have to put parentheses around it and write a 3 out front. Great. Then the 2nd step is that a school orders, one set of wipers and one pair of nozzles for each of its 150 students. So that means that for one student, the school wants one wiper set and two nozzles since they're a pair of nozzles. And we know that one wiper set costs w dollars, so for one student, the school is going to pay w plus the cost for the two nozzles, which we know is 2n. So the school pays w+2n for every student. However, they have 150 students to pay for, so that means that the amount the school will pay is 150(w+2n), so this is our final equation. The updated total future earnings will be equal to 3 times the quantity 84w+60n, and then, added onto that, 150(w+2n) to account for the orders from the school. So looking back at the quiz that you took, I'm sorry, I left out the word total from the quiz, these are the same quantity. So we need to figure out which of these four choices corresponds with or is equal to this expression up here and the correct answer is this 4th choice. Now this doesn't look exactly like this equation, number for number, and that's because the 150 has been distributed to the two terms inside the parentheses. So, if we do that, we keep the first part of the expression the same and then I just multiply 150 by either the terms in the parentheses, just like you did with all of your practice using the distributive property. 150 times w is just 150w and 150 times 2n is 300n. So that's why the final answer is the correct choice. Great. Now, we're one step closer to giving Grant a very nice and easy to use equation for the amount of money he's going to earn.

## Simplifying

The equation that we ended up with at the end of the last quiz for Grant's total future earnings, taking into account the two updates to the number of orders he has received, is definitely helpful. But I think that we can still make it look better. In fact, I know we can because I see that there are some like terms here that have not yet been combined. Using your understanding of the distributive property and just simplifying expressions in general, please simply this expression as much as you can. You'll notice that I actually, in the last quiz, already did part of it for you. So, see if you can remember how to do this.

## Simplifying

Simplifying this expression is really just property. Let's multiply each term inside this 1st set of parentheses by 3. And we want to multiply each term within the we just multiply within each term to simplify. And last, but not least, we add like terms and we end up with a final answer of 402w + 480n. This is interesting. We can see finally, now that everything is simplified, how the number of nozzles that have been ordered compares to the number of wipers that have been ordered. Before, when we just had 84w + there were clearly more wipers than nozzles in demand. But now the situation is reversed. More novels have been ordered than wipers. This is one of the many reasons why simplifying expressions is really, really helpful when we're doing word problems. When we have things simplified down as much as we can, we're able to see what real word situations are reflected by the math.

## Another Challenge

Since you're now an expert on using the distributive property, see if you can use it to multiply these two polynomials together. We have x+2, the whole thing, times x+3, the whole thing. I know that this probably looks a little bit tricky because it looks different from anything you've done before, but it actually uses the same distributive property concept that we been talking about. A, here, just happens to have two terms in it. So, just give it a try. See if you can use the distributive property to solve this problem. Remember, simplify your answer as much as you can.

## Another Challenge

As I suggested in the question video, we can use the distributive property to solve this problem. The difference between this and problems we've done before is that the factor outside of the second parentheses, that we want to multiply everything in here by, is itself a polynomial. However, that doesn't change how we use the distributive property. We're still going to take this entire quantity and multiply it by each of the terms in the second parentheses. So the first thing we're going to have is x+2 times x, and then we'll add to that x+2 times 3. Great, so this looks a little bit more manageable, or like each piece will be. At this point, we're going to use a trick that we've talked about before. We're going to use the commutative property of multiplication. You might wonder, where will we use that here, but notice that if we rearrange the order of these two terms. So instead we'll write x times x+2, and you may be wondering why is that useful, but as you can see, if we rearrange the factors within each term like this, then we have what looks like just a standard distributive property problem, like you've done so many times before. Great. We know how to handle that, and we can simplify within each term, rearrange, and combine like terms. So our answer is x^2+5x+6.

## Showing the Steps

We're going to backtrack, for just a second, to make sure that you understand the steps that are required to multiply two polynomials together. So we're just going to incrementally build up to getting the final answer for this. So for the purposes of this quiz, I want you to act like you're only doing step 1 of solving this problem or of simplifying this expression rather. So for now, please just treat this first factor, the x+5, as a block, so as a single unit, and then distribute it to each of the two terms inside the second set of parentheses and then write out what you would have in the slots over here on the right.

## Showing the Steps

The first thing we do is multiply (x+5) times x. So we'll put x+5 here, and x here. And then the second thing we do is multiply (x+5) times -2. So that is as far as I wanted you to go with this quiz. Awesome!

## Every Term Gets Multiplied

So, here's how far we've gotten. We have our first polynomial factor, x+5, treated as a block, and it is multiplying each of the terms inside the second set of parentheses. The next step is to apply the distributive property, once again. You know that we would rearrange each of these terms so that the factor x+5 comes after the factors that only have one term inside them. And then once we got here we would use the distributive property again, to simplify for each of these terms. So let's take a second and compare what we're doing in this step of the problem to what we did in the very first step. In both cases, we're using the distributive property. We used it the first time, so that we could set ourselves up to do it again. Notice though, which numbers and variables are written here. We have an x and a -2 and then we have an x and a +5. And those are just the same terms that are in the two polynomials up here. The things that we're multiplying together are x and x, x and x, x and 5, x and 5, -2 and x, -2 and x, and distributive property, is just make sure that every term in the first polynomial is multiplied by every term in the second polynomial. Here we've reach another point where we can use a sort of shortcut, so we don't have to do quite as many steps in the problem.

## Finishing the Problem

We'll talk more about that idea in a second. But just for good measure, please write down the final answer for this problem. What is the most simplified form of x + 5 times x - 2 that you can find? In addition to simplifying, please make sure that you write your answer in standard form.

## Finishing the Problem

To continue this problem, we're just going to apply the distributive property to both this part of the expression and this part of the expression. And now we simplify within each term and combine like terms. One quick thing to take note of before I write the final, final answer for this question, let's look at each of these terms. First we have x^2. Now x^2 is just x times x, or x times x. The first terms, in each of our factors, multiplied together. Then we have 5 times x, or the second term of our first factor multiplied by the first term of our second factor. Then we have -2x, which is just -2 times x. The first term from our first factor times the second term of our second factor. And lastly, -2 times 5. Or -10, which is just this factor, 5, times this factor, -2. So you can see that we've taken every combination of pairs of terms from these two factors and multiplied them together in order to get our final answer, which is just x^2 + 3x - 10.

## Every Term with Every Term

In working out the answer to the last quiz, we saw that when you're multiplying up using the distributive property a couple of times, multiplying every term in the first expression by every term in the second expression. So we're taking every possible combination of pairs of terms, using one from the first expression and one from the second expression. This is actually a general rule that applies to multiplying any pair of polynomials, regardless of long it is. So we could use this trick even if there were 10 terms inside the first set of parentheses and 3 terms inside the second set of parentheses, or any other number of terms in either one. In any situation like this we just need to go through one by one, each of the terms inside the first set of parentheses, and multiply them by the terms over here. So in this case, we start with x, and we say x times the first term over here, which is x, so we have x times x. And the second thing we do is multiply x by the next term in the second set of parentheses, x times -2. Then, since we've exhausted all the terms inside the second set of parentheses, we can move to the next factor inside the first set of parentheses, which is 5. 5 applies to the first term in the second set of parentheses, which is x, and then also to the second term, which is -2. So stepping one by one, through the terms inside the first set of parentheses, and applying them each to all of the terms inside the second parentheses, before moving on to the next term, in the first set of parentheses, is a great way to make sure that you have actually covered all of your bases, and made all of the combinations of pairs or terms that you can.

## Multiplying Polynomials

Here's a chance for you to try out how this works using a slightly bigger polynomial. So we have a polynomial of two terms, as our first factor, multiplied by a polynomial of three terms, as our second factor. So remember start with the 3x over here and multiply it by each of the three terms in the second set of parentheses, and then, distribute the 1 to each of these. So all we're really doing is using the distributor property first, only looking at the 3x and then second, only looking at the 1. So, give it a try. If this is difficult that makes perfect sense, this is a really tricky concept to grasp. We will give you plenty of practice to make sure that you understand it.

## Multiplying Polynomials

We know that to multiply this polynomial by this polynomial, we need to ensure that every term over here is multiplied by every term over here. So if we take these two terms one by one, and make sure that each of them is multiplied by every term over in the right=-hand expression, then we'll cover every combination of pairs of terms and also make sure that we don't repeat any. So I'm going to start by distributing the 3x to every term over here. And now that I've multiplied it by each of the three terms that I needed to, I move on to the 1. At this point, all that's left to do is to simplify each term and combine like terms. And we get a final answer of 3x^3-14x^2+x+2.

## The Big One

Lucky you. You get to do another challenge problem right now. Here we have two polynomials once again. This one has 4 terms and this one has 3 terms and I would like you to multiply them together, and of course, simplify as much as possible. I know that you're probably not very pleased with me for asking you to do this, but the reason I am asking you to do it is because I think that this is a great way to make sure that you're actually being very careful with writing out every step in the process for solving this problem. So I know it's not going to be particularly exciting, but if you're careful the first time you do it, then you won't have to do it again. So I'll just remind you. Write everything out, and remember you just want to take each term in the first set of parentheses and multiply it by every term in the second set of parentheses. And then, add all those many, many terms together and simplify. Good luck.

## The Big One

I've given myself a whole bunch of room, so that I can write out all the steps that I'm going to need to simply this expression up here. So as we talked about before, in other problems where we're multiplying polynomials, to deal with these 2, I'm going to start the 1st term of the 1st polynomial and make sure that I multiply it by every single term in the 2nd polynomial. So I'm going to write out all of those terms to start out. So I have 3 terms here, reflecting the fact that I multiplied this term by each of the 3 terms in this expression. Now that the -4x^2 is taken care of, I'll move on to the x, since that's the next term in this 1st polynomial. Once again, I should add on 3 more terms to what I'm building up here, 1 for each of these terms, since I'm multiplying this by all of them. So now I have 6 terms total, which I can double check by counting them all out. And this is actually a really good way to check that you're on the right track, in terms of the spot in the problem you're at. If I've already multiplied out the terms that I need for the first 2 terms of the 4 that I'm going to need to multiply. And there are 3 terms that I'm multiplying each of them by, that'd make sense that I would have 2 times 3, or 6 terms total. Now let's continue, by multiplying all of these by term in our 1st polynomial, the -6, and now we have a super long expression. This looks pretty miserable right now, but hopefully, as we continue to simplify within the terms, then combine like terms, things will get to look a little bit nicer. Really though, in terms of new material that this problem covered, our job is done. All that's left is for us to simplify. Although, of course, this is a particularly complicated simplification process, because there are so many terms. We just have to be super careful. We end up with this lovely, very long final answer of -12x^2y^2+20x^3+3xy^2+6y^3-17x^2-10xy-18y^2+33x+6y-18. Wow. That was really difficult, and if you got this perfectly right, I am so proud of you. I actually made a couple of mistakes myself when I was redoing the problem on the tablet here. You can see actually, when I was doing my work, that I had to put little check marks above all the terms in this step before I got to the final answer. The reason that I did that, is that I wanted to make sure that as I rewrote each term down here, and combined it with it's like terms, that I prevented myself from dealing with it again. So, I put a little check mark above a term, when I wrote it down here, that also let me check that I actually had dealt with every single term that I needed to deal with. I think at this point, if you got this problem right, you have mastered multiplication of polynomials. Congratulations. That's pretty huge.

## Setting the Price

It took Grant a while to process the orders that he had gotten from the school, and from Mr. Belvedere, but in the end, he finally had a moment to relax. He had visions of little children running around with glasses, wipers, and nozzles, and of course, of his new friend. What a nice old man. He felt so excited and so satisfied that, pretty soon, his products would be keeping the glasses of people all over the world nice and clean, but he realized he'd left out one major detail. He hadn't yet set the price for his wipers or for his nozzles. In terms of the equation that we came up with for his total future earnings, he hadn't picked a value for w yet, or one for n. Remember that w is equal the price per wiper blade set and n is just equal to the price per nozzle. Grant thought long and hard about how much he wanted to charge for each wiper blade set and for each nozzle. But finally, he came to a decision, he set the price per wiper blade set at \$30 and he set the price per nozzle at \$12. Now, these might look like kind of steep prices, but, hey, Grant had worked really hard on both of his products and they're actually pretty intricate designs. Takes a lot of work to make nozzles and wipers this small that aren't just going to break their first use.

## Future Earnings

Now that we finally have values for w and n to equal, we can plug them into our equation for total future earnings, using substitution, like we have so many times before. Can you use these two numbers, 30 and 12, in the proper places in this equation to come up with a number for the total amount of money that Grant will make off of the orders that have come in so far? Think about what numbers you want to substitute where, and then be sure to be careful about order of operations.

## Future Earnings

I rearranged things on the screen here a little bit. just so I'd have room to write my answer out, but this is all the information that you had to start the last quiz with. Remember, from our practice using substitution before, that if we see a variable in one equation, but we know that that variable is = to some other expression, then we can substitute in that alternate expression in the place of the original variable. So, in the case of w and n, for example, we have these two numbers that we can now replace them with. So, we see w here, 402w, and w here, w = 30. So I can take the 30, and move it into this slot. And I can do the same thing with n and 12. I'm going to rewrite this side of the equation again, but this time with our new numbers plugged in. In place of w I put 30 and in place of n I put 12. Now we need to recall our understanding of order of operations. So PEMDAS. The thing we need to decide here is whether we multiply or we add first. Since we have we have two terms that are multiplied and another two terms that are multiplied but then also an addition sign. In our past practice with simplifying expressions we always simplified within each term, or multiplied everything out, before we combined like terms or added, and that's completely in line with PEMDAS. Remember, this is an acronym to help you remember the order in which you do different mathematical operations within a given expression. So the M here, which stands for multiplication, comes before the A, which stands for addition. So we're going to do our multiplication first. 402 times 30 = least, we add these two numbers together. This gives us a final answer of 17,820. Much to the delight of Grant and his venture capital funders. They're going to bring in \$17,820 with just the orders that have been placed so far in the first few weeks. That is really incredible. Grant has proven that he has a product, or actually two products, that are popular and people think are really going to be useful. He is convinced he's out to change the world, or at least the way that glasses wearers see the world. Congratulations on great work, we covered a ton of material, and I am sure that you did great.