Learn about Algebraic Long Division Video

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Learn about Algebraic Long Division

Learn about Algebraic Long Division I’ve been asked to make a video on how to break division or how to break a long division. So I’ll make a video on Algebraic long division. So let say we wanted on the screen to make of a problem and let say we wanted to divide. We wanted to see how many times does, I start with a fairly straight forward problem. How many times does 2x+1 go into, I don’t know, let say it’s 8x^3-7x^2+10x-5. So what we do is we just take, actually just as exact same way that you would do with long division – traditional long division of multiple digit. You would and the 2x+1x and you look what is the highest degree term and that’s really are we going to pay attention to do most of the time. So, the first step is you see and get the highest degree term is 2x. How many times does 2x go into the highest degree term of the number or not the number, the expression that we are dividing into. So you say how many times does 2x go into 8x^3? Or we could do a long division on the side but you can imagine of entry that this pretty straight forward. So if you have 8x^3 divided by 2x, that is equal to 4x^2, right? So 2x goes into 8x^3=4x^2 and this is the key that you don’t want to write the 4x^2 here. And you want to keep everything in the correct places, so when you are dividing numbers. You think of the ones, tens, the hundreds, and the thousands place and etc. When you dividing polynomials, you can go to think of the x^0 space, the x^1 space through the x space and the x^2 space, the x^3 space. So when we say that 2x goes into 8x^3 of 4x^2 times all just right that in the x^2 spot. It goes into 4x^2 times. Now we take that 4x^2 and we multiply it by expression. I think you are already see that this is very similar to long division. And actually if x was a ten, it would be identical to long division. And I let you think about that, right? If x was ten and this would be the thousands place that would be 8000 minus of that you have negative digits which doesn’t make bunch through sense but I think you got what I’m saying. But anyway, we back to this Algebraic long division although I think it’s very important to see the parallels between this and traditional long division. Why we said that 2x goes into 8x^3 of 4x^2 times. Now what we can is we can multiply 4x^2*2x+1. So 4x^2*1, that’s 4x^2 so we can write that in the x^2 position place and we can write it for x^2. And 4x^2*2x=8x^3, 8x^3 and this is the + here and now it just like we do with the traditional long division. We can subtract this from this. So -7x^2-4x^2=-11x^2 and then 8x^3-x^3, so we can ignore that right there. And if we want we can bring down the rest of the number but maybe just for fun it will bring down the next but it just like we do in traditional long division. And if you ask about me erase this over here. And when you erase this because I think it will might find this that real state usual, alright I’m done. And actually it doesn’t hear to bring down the whole thing. Just solve if you understand of what we’re doing. We’re saying if you were to divide 2x+1 to this entire expression and you say that it goes in 4x^2 times. Now are that you can go to call it our intermediate remainder is what’s left, right? This is what’s left and you can almost imagine 4x^2*2x+1, this is 8x^3+4x^2+0+0, right? Because it doesn’t contribute anything to this spot but anyone is left over is this expression. If you take this minus this whole expression and you get what’s left over. Now we just do the same thing over. How many times do 2x, we just look at the highest order term. How many times does 2x go into -11x^2? So let do something jusdt right at there on this side again. Actually, we do it here. So if we were to take -11x^2 divided by 2x and that is equal to what? That is equal to -11/2x, right, so 2x goes into -11x^2-11/2x times. So we will write that in our x place, so -11/2, we can write that as 5.5 of that all just as fraction 11/2x and now what is -11/2x*2x+1. So -11/2x*1 is -11/2x and we want to write that in the x position. I’ll switch colors just to not be monotonous, so -11/2x*1=-11/2x and then -11/2x*2x or we should that is but you can multiply them all and it will be -11x^2. And I think you see what we are doing after every step were canceling out the largest degree of the polynomial where dividing into. For now let subtract this expression from this and we will get kind of our new intermediate remainder and maybe it will be the four remainder. So let say -11x^2-11x^2=0, so we don’t have write anything there. 10x-(-11/2x), remember were subtracting this negative n umber from 10x. Does you subtracting a negative number is like adding a positive number, right. So you can view this as 10+11/2, right? So 10+11/2, that is 20/2+11/2=31/2 or 15.5 or just 31/2x. And then you can say that there was a zero here and when you subtract the zero from -5 and you get -5. And now that we say, how many times does 2x go into 31/2x? Let’s do a little work on this side here. So if I have 31/2x divided by 2x, well, the x it will just cancelled out, right? And so this is the same thing as this equal to 31/4, right. This is the same thing as 31/2*1/2, so it’s 31/4. So 2x goes into this expression of 31/4 times and I’ll switch colors, I’ll switch to green. So and that’s a positive, right. You are doing a positive to positive, so +31/4 times and I’m writing that in the, you can view that in the constant space from the x to the zeros space or the one space and see them. So it goes into 31/4 times, 31/4*1=31/4 and 31/4*2x=31/2x, right? And now we subtract and this is a plus here. We subtract the green expression from the light blue expression. And we’re left with this and we just expect this from this and you left with zeros with nothings show up there. And we are left with -5-31/4 and we just do a little of fraction work here and that’s equal to – see -5/4-20-31, all of that are over four. So minus, so that is equal to what? -20, that’s equal to -51/4. So our answer is 2x+1 goes into 8x^3-7x^2+10x-5, it goes into 4x^2-11/2x+31/4 but there is a remainder and this is the remainder. And so I wait of visualizes, so in other way to think about this problem and just so that we don’t – so it actually useful and when we actually solving real problems. And that we just don’t view this of some kind of mechanical way to get the problems right on a tests that only tests algebraic long division. As another way to write this relationship, you can write that and we do it in another. And see that I have used and to have many colors already. So you could write that 2x+1*4x^2, that’s in x-11/2x+31/4+the remainder. So when you multiply this two out and then if you were to add the remainder 51/4 that would equal and then let me draw a dividing line when it confuse you with all the stuff here and that would equal to this. That would equal to 8x^3-7x^2+10x-5. Anyway that I hope it helps. See you in the next video.

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Learn about Algebraic Long Division

I’ve been asked to make a video on how to break division or how to break a long division. So I’ll make a video on Algebraic long division. So let say we wanted on the screen to make of a problem and let say we wanted to divide. We wanted to see how many times does, I start with a fairly straight forward problem. How many times does 2x+1 go into, I don’t know, let say it’s 8x^3-7x^2+10x-5. So what we do is we just take, actually just as exact same way that you would do with long division – traditional long division of multiple digit. You would and the 2x+1x and you look what is the highest degree term and that’s really are we going to pay attention to do most of the time.

So, the first step is you see and get the highest degree term is 2x. How many times does 2x go into the highest degree term of the number or not the number, the expression that we are dividing into. So you say how many times does 2x go into 8x^3? Or we could do a long division on the side but you can imagine of entry that this pretty straight forward. So if you have 8x^3 divided by 2x, that is equal to 4x^2, right?

So 2x goes into 8x^3=4x^2 and this is the key that you don’t want to write the 4x^2 here. And you want to keep everything in the correct places, so when you are dividing numbers. You think of the ones, tens, the hundreds, and the thousands place and etc. When you dividing polynomials, you can go to think of the x^0 space, the x^1 space through the x space and the x^2 space, the x^3 space. So when we say that 2x goes into 8x^3 of 4x^2 times all just right that in the x^2 spot. It goes into 4x^2 times. Now we take that 4x^2 and we multiply it by expression. I think you are already see that this is very similar to long division. And actually if x was a ten, it would be identical to long division. And I let you think about that, right?

If x was ten and this would be the thousands place that would be 8000 minus of that you have negative digits which doesn’t make bunch through sense but I think you got what I’m saying. But anyway, we back to this Algebraic long division although I think it’s very important to see the parallels between this and traditional long division. Why we said that 2x goes into 8x^3 of 4x^2 times. Now what we can is we can multiply 4x^2*2x+1. So 4x^2*1, that’s 4x^2 so we can write that in the x^2 position place and we can write it for x^2. And 4x^2*2x=8x^3, 8x^3 and this is the + here and now it just like we do with the traditional long division. We can subtract this from this. So -7x^2-4x^2=-11x^2 and then 8x^3-x^3, so we can ignore that right there. And if we want we can bring down the rest of the number but maybe just for fun it will bring down the next but it just like we do in traditional long division.

And if you ask about me erase this over here. And when you erase this because I think it will might find this that real state usual, alright I’m done. And actually it doesn’t hear to bring down the whole thing. Just solve if you understand of what we’re doing. We’re saying if you were to divide 2x+1 to this entire expression and you say that it goes in 4x^2 times. Now are that you can go to call it our intermediate remainder is what’s left, right? This is what’s left and you can almost imagine 4x^2*2x+1, this is 8x^3+4x^2+0+0, right? Because it doesn’t contribute anything to this spot but anyone is left over is this expression. If you take this minus this whole expression and you get what’s left over.

Now we just do the same thing over. How many times do 2x, we just look at the highest order term. How many times does 2x go into -11x^2? So let do something jusdt right at there on this side again. Actually, we do it here. So if we were to take -11x^2 divided by 2x and that is equal to what? That is equal to -11/2x, right, so 2x goes into -11x^2-11/2x times. So we will write that in our x place, so -11/2, we can write that as 5.5 of that all just as fraction 11/2x and now what is -11/2x*2x+1. So -11/2x*1 is -11/2x and we want to write that in the x position. I’ll switch colors just to not be monotonous, so -11/2x*1=-11/2x and then -11/2x*2x or we should that is but you can multiply them all and it will be -11x^2.

And I think you see what we are doing after every step were canceling out the largest degree of the polynomial where dividing into. For now let subtract this expression from this and we will get kind of our new intermediate remainder and maybe it will be the four remainder. So let say -11x^2-11x^2=0, so we don’t have write anything there. 10x-(-11/2x), remember were subtracting this negative n umber from 10x. Does you subtracting a negative number is like adding a positive number, right. So you can view this as 10+11/2, right? So 10+11/2, that is 20/2+11/2=31/2 or 15.5 or just 31/2x. And then you can say that there was a zero here and when you subtract the zero from -5 and you get -5.

And now that we say, how many times does 2x go into 31/2x? Let’s do a little work on this side here. So if I have 31/2x divided by 2x, well, the x it will just cancelled out, right? And so this is the same thing as this equal to 31/4, right. This is the same thing as 31/2*1/2, so it’s 31/4. So 2x goes into this expression of 31/4 times and I’ll switch colors, I’ll switch to green. So and that’s a positive, right. You are doing a positive to positive, so +31/4 times and I’m writing that in the, you can view that in the constant space from the x to the zeros space or the one space and see them.

So it goes into 31/4 times, 31/4*1=31/4 and 31/4*2x=31/2x, right? And now we subtract and this is a plus here. We subtract the green expression from the light blue expression. And we’re left with this and we just expect this from this and you left with zeros with nothings show up there. And we are left with -5-31/4 and we just do a little of fraction work here and that’s equal to – see -5/4-20-31, all of that are over four. So minus, so that is equal to what? -20, that’s equal to -51/4.

So our answer is 2x+1 goes into 8x^3-7x^2+10x-5, it goes into 4x^2-11/2x+31/4 but there is a remainder and this is the remainder. And so I wait of visualizes, so in other way to think about this problem and just so that we don’t – so it actually useful and when we actually solving real problems. And that we just don’t view this of some kind of mechanical way to get the problems right on a tests that only tests algebraic long division. As another way to write this relationship, you can write that and we do it in another. And see that I have used and to have many colors already. So you could write that 2x+1*4x^2, that’s in x-11/2x+31/4+the remainder. So when you multiply this two out and then if you were to add the remainder 51/4 that would equal and then let me draw a dividing line when it confuse you with all the stuff here and that would equal to this.

That would equal to 8x^3-7x^2+10x-5. Anyway that I hope it helps. See you in the next video.

Transcription by: Scribe4you Transcription Services