Algebra 1 – Module 3, Topic 2 – Adding and Subtracting Polynomials Transcription
Hi guys, welcome to Algebra I. Today's topic is going to focus on adding and subtracting polynomials. All that you know about simplifying expressions is really going to take you far on this topic. You ready to get started? Let's go.
Let me take you back a bit to the distributive property. You're going to apply it when you're working with polynomials, when you have to add and subtract them, you end up having to apply this a bit. Just to review, you remember the distributive property. If I see that 2 out on the left, I can distribute it throughout what's in my parentheses and I bet somebody showed you these little arrows to show what you were doing. I can say I need to multiply 2 times 4x, which is 8x, and I need to multiply 2 times +1, which is 2. You could say 2 times that quantity, 4x + 1, after I had applied the distributive property, I could write that as 8x + 2. Let's apply that property a few more times, get it in our system. 
This expression, 7 + 3(x-2), for this one I'm going to work down a bit; I think that might help this make a little more sense to you. Let's move that over slightly. The 7 is just going to go along for the ride for a bit. Let me erase that dot so you don't think it's subtraction. 7 is just going along for the ride for a second. 7 +, now I do need to apply my distributive property here, for the second bit, so 3 times x, that's 3x, and 3 times -2, because that's how I interpret that, 3 times -2, that's -6, so you'd write it as a -6. Then you look back at your problem and you can combine some like terms here. You do have your 3x, which doesn't have anything else like it in that expression, so there's nothing to combine that with, so it's now just coming on down here. I have two constant terms that I can combine, to write it more simply. 7 - 6, that's 1, so that means I can write that as a +1. You apply the distributive property and you simplify that original expression to 3x + 1. 
Let's keep going. This one, let's do the same thing. Let's move that over a little bit. I have 5 - (3x + 4). That 5 is just going to go along for the ride, for a minute. When you see something like this, I hope you remember, you have this minus and then this 3x + 4 in parentheses. Treat that as if there is a 1 right in front of it. What's really happening is you're distributing a -1 throughout what's in your parentheses. -1 x 3x, that's a -3x, and the -1 x 4, that's a -4. Now I'm at the point where I see I have like terms I can combine. This -3x doesn't have anything like it. It's the only term in this expression with an x, so I know that this is -3x. That's just going to bring on down. I do have this 5 - 4, so 5 - 4, I know that's 1, and it's a positive 1, so I can write that as +1. That original expression, it simplified to -3x + 1. Jogged your memory a bit with the distributive property? 
Let's get into the polynomials. There are three steps to follow when you're adding and subtracting polynomials. You first want to rewrite the problem, then you want to combine like terms, then you want to make sure your final answer is written in standard form. Remember, standard form, that's when you have your exponents, when they go in descending order, so the leading term has the largest exponent and then you just go in descending order from there. Your final answer always has to be in that form. While we're working with these problems, keep these three steps in mind. We're going to rewrite it, we're going to combine like terms, then we're just going to make sure we're in standard form.
Let's look at the first one. Deep breath, because I know it looks massive. There's a lot going on, but it's really going to break right on down to something that looks a lot more simple. Our first polynomial, it's actually a trinomial, we've got three terms. 7x3 + 6x2 + 2x and we're going to add that to 3x3 - 4x2 + 7x - 5. We're going to start by rewriting the problem. When you're adding polynomials, rewriting the problem is essentially just getting rid of the parentheses. When you're adding, the rewriting is just break those terms out of the parentheses. I have 7x3 + 6x2 + 2x + 3x3 - 4x2 + 7x - 5. I just broke that problem out of the parentheses. 
Now what I'm going to do, just so this doesn't get too crowded here, is I'm going to get rid of that problem. Now we've broken it out of the parentheses, so let's just look at it by itself. I think it's going to help us look at this a little better. Let's click off of it so I can write. Here we go. Now we're at step two, where we need to combine like terms. How I find out what terms are alike is the terms have to have the same variable raised to the same power. A coefficient doesn't need to be the same, just the same variable raised to the same power. Those are our like terms. Since we know that we need our answer to be in standard form, let's just start out with the largest exponent, so that we know that when we're at the end, more than likely we've got it written in the right form. Scan your terms and you notice that you have a 7x3 and you have a 3x3. Let's take those two terms first.
I've got the 7x3 and I've got this 3x3. You notice how I took the sign out front, because it tells me what the sign of that term. I'm going to come off and do some scratch work over here. 7x3 + 3x3, here we're just adding like terms. We're not multiplying anything together, so we're not going to change the value of our exponents because we're not finding any products here, we're combining like terms. 7x3 + 3x3, that's just 10x3. Essentially all you're doing is adding your coefficients and just carrying along that variable to that power. 
I know we've got the first set handled, that 10x3, so that's out of the way. Go in descending order, so now let's look for exponents of two and I notice I have this 6x2 and I have this -4x2. Let's combine those like terms. 6x2 and I'm combining it with a -4x2, so I'm just going to treat that like subtraction. 6x2 - 4x2, so remember, you're not really messing with your exponents at all, you're really just focused on the coefficients. 6x2 - 4x2, well, 6 - 4, that's just 2, so this is 2x2. Now you've handled this. You've handled the exponent of 3; you've handled the exponent of 2, so let's look for exponents of 1, which you really don't see. 
I'm going to start squaring things now, not as exponents, but just putting them in boxes to help me focus. I have a 2x2 and I have a 2x and I have a +7x, so I'm going to combine those like terms together. 2x + 7x, so essentially I'm just combining like terms here. 2 + 7, that's 9, so this is 9x. I'll combine those 2x + 7x, that's 9x. That's taken care of. All that's left is that lonely little -5. That's our constant term; I'm just going to write it down here. 
Now we're at the point where we can write our answer and just make sure it's in standard form. I'm going to bring all these little individual answers we got together. I'm going to write this as ... Let's see the next thing to erase, give ourselves some room. Let's get rid of our problem because now we have the answer and I'll write it right up here. Let's get that out of the way. When we added those polynomials together, it all really simplifies to 10x3 + 2x2 + 9x - 5 and that is our answer. That is what we get when we add those two polynomials together. Then you can double check and we do have it in standard form. Our exponents are going from largest and then in descending order. You ready to try another one? Let's do it.
10a3 - 3a2 + 4 and we're going to add that to -6a3 + a2 + 1. Let's bring that up a little bit. You remember step one is rewrite the problem and when we're adding, essentially that's just breaking the terms out of the parentheses. 10a3 - 3a2 + 4, now look here. You see we have a plus this -6a3? We don't like to have the signs bump into each other like that. Instead of writing +-6a3, I'm just going to write -6a3. Didn't change the meaning at all because adding and negative means the same thing as subtraction. Remember that from pre-algebra. I can just rewrite +-6a3 as just -6a3. Then just break the other terms out of parentheses, a2 + 1. Now we're at the point where we can combine like terms, so I'm going to give us a little more room up here. Let's get rid of our problem and let's bring this up. We can get some space to work. 
Now look for your like terms you can combine. Essentially you're looking for the same variable rates of the same power, to identify your like terms. I see I have this 10a3 and I have this -6a3, so I can combine this one and this one. I'll come off and do a little scratch work. 10a3 - 6a3, basically you're just subtracting your coefficients, here. 10a3 - 6a3, that's 4a3, because 10 - 6 is 4. You handled that. Looking for your exponents in descending order, so you scan your terms and you see you have this -3a2 and this +a2, so those are your next two that you're going to want to combine. -3a2 +a2 ... You remember, when you don't see a coefficient, there's an invisible 1. -3a2 + 1a2, that's just -2a2, so you've handled those square terms. Now I see I have two constants left. I have this +4 and I have this +1, so 4 + 1, that's 5. You've combined all the terms you can combine, so you're ready to write your answer. 
I do have some space over here, so I'm going to write it right here. Essentially just bring all those individual answers together, since we kind of worked out our problem in chunks. 4a3 - 2a2 + 5, because that 5 down there is positive, and you are all done because we have it in standard form. Starting to feel a little better about it? Let's try another one. It usually takes three to feel like, "Okay, I think I got it, I think I'm ready to try one."
Here we have 8b4 - 5b2 + b and we're adding that to 3b2 + 5b. You remember step one? Rewrite the problem. When we're adding, that's essentially just breaking it out of parentheses. 8b4 - 5b2 + b, we're going to add that to 3b2 + 5b. Now we're at step two, where we can combine like terms. This is where I get us some more space to work. Let's get that out of the way and let's bring that up a bit. Now let's get to work. I scan the problem and because I know I want my answer in standard form, I'm always going to look for the largest exponent first, to start identifying where my like terms are. I see this first term has an exponent of 4. It's the largest one, but I notice there are no other terms that have an exponent of 4. There's nothing to combine that term with, it just keeps going along for the ride. 8b4, just continued on being 8b4, it didn't combine with anything. 
Keep it in mind that we're trying to go in descending order with our exponents. We look for powers of 3. We don't see any powers of 3. We look for powers of 2 and I do see a couple terms that have a power of 2. I see this -5b2 and I see this +3b2. I can combine those. -5b2 + 3b2 ... Remember, essentially we're just focusing on the coefficients, so -5 + 3, that's -2, so this is -2b2. We've handled that. I see all we're left with here is two terms that both have the variable b and those are our last pair of like terms to combine. We have this b + 5b, so b + 5b ... Remember, when you don't see a coefficient, there's an invisible 1, so 1b + 5b, that is 6b. Look back at the polynomial. There's nothing else to combine.
Now we're at the part where we can write our answer. I'm going to write it up at the top, so let me get that out of our way, then we can get the answer up here. Let's bring our individual answers all together, because we solved our problem in chunks. 8b4 - 2b2 + 6b. It's in standard form because the exponents are going in descending order. All done. You ready to try one? Okay, how about two? Go ahead and press pause, take a few minutes. Take your time and work through these problems. Take it in chunks, just like we did. It probably is a little more easier to manage that way. When you're ready to check your answers against mine, go ahead and press play again. 
You ready to check? Let's check these. That first one, 11y3 + 3y - 2 + 4y2 + y + 5. The answer there, you should've got 15y2 + 4y - 3. On the second one, we had -5x3 + 7x - 1 and we were adding that to 8x3 + 3x2 + 2x + 2 and the answer, you should've got 3x3 + 3x2 + 9x + 1. If you want to see how I did that, keep on with me. 
Remember step one, when you're adding, essentially you're just breaking those terms out of the parentheses, is the first step. 11y2 + 3y - 2 + 4y2 + y + 5. Now that I've got it broken out of parentheses, give us some room to work. I can get rid of that, I'm going to move this up a bit. Now you're at the point where you need to scan your terms because you've got to find the like terms because you've got to combine. 
Because we know we want our answer to be in standard from, just go ahead and start looking for the terms that have the largest exponent first. I see my largest exponent in this polynomial is 2 and the terms that have that exponent are 11y2 and then this +4y2, so I do some scratch work 11y2 + 4y2, that's 15y2. We've handled this. Now I see I have 3y + 1y, so 3y + 1y, because there's an invisible 1 that I don't see. 3y + 1y, that's 4y, so I've handled those terms. I have two constant terms left. I have this -2 and this +5. -2 + 5, that's a positive 3. Those are my last two terms that I needed to combine. Now you're at the point where you can write the answer. Since we worked the problem in chunks, we're going to take all our chunks and put them together. 15y2 + 4y + 3, that's how I got that first one. Need to see the next one?
Going to bring that up a bit. Here we go. -5x3 + 7x - 1 + 8x3 + 3x2 + 2x + 2, so remember for adding, essentially your first step is to break it out of parentheses. -5x3 + 7x - 1 + 8x3 + 3x2 + 2x + 2, so I've broken it out of parentheses and just to help us focus, I'm going to get rid of that original problem and just zone in on this one. Now I'm at the point where I'm looking for my like terms to combine and because I know I want my answer in standard form, I'm going to start with my largest exponents first. I see that I have -5x3 and this +8x3, so -5x3 + 8x3, that's 3x3. You've handled those terms. Now I'm going in descending order, so I'm looking for x2 and there's only one little term that has an x2, so it just comes along for the ride, there's nothing to combine it with. I know 3x2 is just going to be around.
Now I see I have 7x + 2x, so 7x + 2x, that equals 9x. I handled that. Now I have two constant terms left. I have this -1 combined with this +2, so -1 + 2, that's a positive 1. Now we have all our individual chunks and we just need to bring it together for our final answer. I think I left myself enough space, I can write it right over here, to the right. 3x3 + 3x2 + 9x + 1, and that's how we got that answer. You've handled addition, now let's go to subtraction. I want you to think back to one of those early problems that we did in this lesson, where we handled this problem with the distributive property, so to jog your memory, remember what we did on this one. We said we have a five our front, it just stayed along for the ride, and we had to subtract that quantity in the parentheses, so we said it's minus this, there's a 1 here I don't see, so essentially I'm distributing this -1 throughout this quantity in here. -1 x 3x, that's -3x, -1 x 4, that was a -4. 
We said okay, now we need to combine like terms. The -3x didn't have anything alike, so it just came along for the ride, that term right there, we've handled it. Then we had this 5 - 4 and 5 - 4 is 1, so +1. Remember how we did this? The part that I really want you to focus in on this is that it was a subtraction type problem. It was 5 - that 3x + 4 and I want you to pay attention to what we had to do with distributing that -1. We have to use that same process when we subtract polynomials. Look at this one. 
We have 8x2 + 5a - 4a2 + 2a- 6. Let's give ourselves a little room, I'm going to bring that up a bit. You remember when we're dealing with polynomials and we're adding and subtracting them, the first step is to rewrite the problem. Now when we're adding, rewriting is essentially just breaking those terms out of the parentheses, but when we're subtracting, it's not just that. We have to distribute this -1, that's kind of invisible, that we're kind of not seeing. That's the extra step that's involved when you're subtracting polynomials. 
I look at that first binomial that I'm just going to break out of the parentheses; nothing's really going on with that one. As far as this trinomial was concerned, this second batch, there is a -1 here that I don't see, it's kind of invisible. I need to distribute this -1 to each of these terms before I can go any further with this problem. -1 x 4a2, that is ... Let's show the work, just so it makes sense. -1 x 4a2, that's -4a2. That first term I -4a2. Now we need to multiply that -1 x 2a, so -1 x2a, that's -2a, so -2a. Now I need to distribute that -1 to that -6, so -1 x -6, that's a +6, so +6. Now that I've rewritten the problem and because it was subtraction, it involved me distributing that -1 throughout, now I can continue on to simplify this expression.
Now it's pretty much the same as when we were adding. I think I left myself enough room, I can work right here. I see I have this 8a2 - 4a2, so let me ... I know what I can do. I will get rid of this bit and show that work right there. 8a2 - 4a2, that is 4a2. I've handled this. Now I have 5a, and it's positive, -2a, so 5a - 2a, that's 3a. You've handled those terms. All you have left now is that constant term, that 6. It just goes along for the ride, you've got a 6. Now we're at the point, because we solved our problem in chunks, we just need to bring all our chunks together and then we'll be at our answer. I would have 4a2 + 3a + 6, and you are all done. Subtraction isn't too much unlike addition; you just have that extra step where you have to distribute that -1 throughout that second set of terms. 
Let's try another one. 7y4 + 3y2 - 1 - 6y4 + 4y2 - 2y + 1. Step one, rewrite the problem and because we are subtracting, keep in mind we're going to have to use the distributive property for the second set. For the first set, just break it out of parentheses. 7y4 + 3y2 - 1, so now that I'm at my second set, remember there is an invisible 1 here. There is a -1, because you're subtracting, that you have to distribute throughout those four terms. You have -1 x 6y4, now come over here, so -1 x 6y4, that's -6y4. -6y4 and I have -1 x 4y2, so -1 x 4y2, that's -4y2. -4y2 and I can see things are about to run into each other here, so let's just bring that down a little bit. Then I need to distribute this -1 to that -2y, so -1 x -2y, that's a +2y, so +2y. Then I have this -1 x +1, so -1 x +1, that's -1. I have a -1 right here. 
Now we're at the point where we can combine like terms. I'm going to get that last bit of work out of our way because we're going to need some extra spaces to get a little scratch work in. Let's see. I know I want to go in descending order of exponents, so which exponent is my largest in this case? I'm scanning the terms, it's 4. I have a 7y4 and I have a -6y4, so 7y4 - 6y4, that is 1y4, and I don't have to write that 1. I've handled that. I don't have any y3 terms, but I do have y2 terms. I see I have a 3y2 right here and I have a -4y2 there. Let's combine those. 3y2 - 4y2, so 3y2 - 4y2, that's -1y2. Now I've handled those. I see I do have a 2y term, so that just is going along for the ride. I can squeeze it in right here. 2y is just carrying along. 
I do have a couple of constant terms here. I see I have this -1 and another -1. Let's get some things out of our way here. This problem got large. Let's get this out of the way. Let's do a little erasing and let's bring some stuff up a bit. Let's bring that up and let's bring the work up. Now we've got some more room for scratch work. We were doing -1 - 1, so -1 - 1, that's -2. Now we have combined all the terms that we can, so we're at the point we can write our final answer. Y4 - y2 + 2y - 2. It's in standard form, we are all done. I think you've got one more in you. Let's try another one before you try them on your own. 
-5x3 - x2 + 3x - -4x3 + 4x2. Let's bring that up a bit and let's get to work. The first polynomial, essentially I'm just breaking it out of parentheses. There's no multiplication going on out here that's telling me I need to do anything else. -5x3 - x2 + 3x, okay. Now I'm subtracting, so it's like I'm distributing a -1 throughout these terms. I need -1 x -4x3, so - 1 x -4x3, that's a +4x3, so +4x3, then I have this -1 x 4x2, so -1 x 4x2, that's -4x2, so -4x2. We've gotten everything out of the parentheses, we did our distribution, so now we're at the point we can start combining like terms. Let's get this old work out of the way, make room for some new scratch work. 
I know I want to go in descending order of exponents. My largest exponent is 3 in this problem, so I have a -5x3 + 4x3, so -5x3 + 4x3, that's -1x3. I've handled that. I noticed I have some x2 terms, so -x2 - 4x2, so -x2 - 4x2, that's a -5x2. You've handled the x2 terms, then all that's left is that 3x. There's nothing to combine with it, so it's just going along for the ride. Now we've handled everything we can as far as combining like terms, so we're at the point we can write our final answer. We would have -x3 - 5x2 + 3x. Got it in standard form, so we are all done. Why don't you try a couple? Go ahead and press pause and take a few minutes, then really take your time and work your way through these problems. When you're ready to compare your answers with me, go ahead and press play.
Ready to check? Let's go. For that first one, -6c5 + 5c3 + c - 4c5 + c3 - 8c. That one was -10c5 + 4c3 + 9c. That should've been your answer for the first one. 5x3 + 6x2 - 1 - -2x3 + x2 + 2, that one, 7x3 + 5x2 - 3. If you want to see how I did those, for the first trinomial, essentially we just need to get it out of the parentheses because there's no multiplication of anything going on out here that I need to handle. I would have -6c5 + 5c3 + c, and because I'm subtracting, there's a -1 out here that I need to distribute throughout those terms. I need to do -1 x 4c5, -1x c3, -1 x -8c. I'm going to come off to the side and do that scratch work. -1 x 4c5, that's -4c5, so -4c5. Then I have -1 x c3, so -1 x c3, that's -c3, so -c3. Then I have -1 x -8c, so -1 x -8c, and that's a +8c. That means I have a +8c. 
If you notice, when you're subtracting, essentially those terms right there, where I'm distributing that -1 throughout, their signs just become the opposite of whatever they were to start with. If you started to notice that pattern, that when you distribute the -1 it changes all the signs in that second group, you can go ahead and jump to that step. That's fine. Algebra is all about the patterns that lead us to these shortcuts, but if you want to continue and do that distribution and really show the steps, that's fine too, because it will get you really comfortable with exactly what we're doing here, with this process. Keep that in mind and let's keep going. 
We're at the point now where we can combine like terms, so let me get us a little more room here to work. Let's get that out of the way, let's erase this right here, let's get rid of that, and let's move this up. Now we've got some room to work here. I know I want my answer in standard form, so scan your terms, you're looking for the largest exponent first, and I see in this problem my largest exponent's a 5. I'm going to combine -6c5 with -4c5, so -6c5 - 4c5, that's -10c5. I've handled those. I don't have any c4 terms, but I do have some c3 terms, so 5c3 and I see I have this -c3, so 5c3 - c3, that's 4c3. We handled the c3 terms. I see I only have two terms left and they're alike, they're both c, or they both have a variable for c. I have this +c and I have this +8c, they're both positive, so c + 8c, that's 9c. I've handled all the terms in that polynomial, so now I'm at the point where I can write my answer in standard form. I'd have -10c5 + 4c3 + 9c and you are all done with that one.
Let's look at how I did that next one. For this one, let's move it up a bit. What you want to do first, check out that first polynomial. There's no multiplication or anything going on out front, so just break it out of parentheses. 5x3 + 6x2 - 1. Now I'm subtracting, so remember you're distributing this -1 throughout these terms. I need to multiply each of those terms by -1. I'm going to come off to the side and do a little scratch work. For the first time, -1 x -2x3, so -1 x -2x3, that's a positive 2x3, so +2x3. Now I have this -1 x x2, so -1 x x2, that's a -x2, so -x2 for that bit. Then I have a -1 x this +2, so -1 x 2 and that's -2. Come up here, I have my -2, then we said you might have noticed it just basically change all the signs in that second bit when you distribute that -1, so maybe you took that shortcut or maybe you did all the work like I did, either way is fine. 
Now we've reached the point where we can combine like terms, so let's give ourselves a little more room to work up here. Let's get rid of that and let's get rid of this and get that out of our way and move this up. Let's start combining. I know I want my answer in standard form, so I'm going to look for the largest exponent first. I see I have 5x3, 2x3, so I can combine those. 5x3 + 2x3, that's 7x3. You've handled those two terms. Now you're scanning again. You see you have some x2 terms. I have 6x2 and -x2, so 6x2 - x2, that's 5x2. You've handled the x2 terms, then you look again and you just have two constant terms left to combine. I have this -1 and that -2. -1 - 2, that's -3. Now you're at the part where we can put each of our little individual chunks together and write our final answer in standard form. 7x3 + 5x2 - 3 and you are all done. 
I hope you really got a grip on how to add and subtract polynomials and I hope you saw how the distributive property really helps you with those subtraction problems. Hope to see you back here soon for some more algebra I. Bye, guys.