In this video, we’re going to

learn to divide polynomials, and sometimes this is called

algebraic long division. But you’ll see what

I’m talking about when we do a few examples. Let’s say I just want to

divide 2x plus 4 and divide it by 2. We’re not really changing

the value. We’re just changing how we’re

going to express the value. So we already know how

to simplify this. We’ve done this in the past. We

could divide the numerator and the denominator by

2, and then this would be equal to what? This would be equal to x plus

2– let me write it this way– it would be equal to, if

you divide this by 2, it becomes an x. You divide the 4 by

2, it becomes a 2. If you divide the 2

by 2, you get a 1. So this is equal to x plus

2, which is pretty straightforward, I think. The other way is you could have

factored a 2 out of here, and then those would

have canceled out. But I’ll also show you how to

do it using algebraic long division, which is a bit of

overkill for this problem. But I just want to show you that

it’s not fundamentally anything new. It’s just a different way of

doing things, but it’s useful for more complicated problems. So you could have also written

this as 2 goes into 2x plus 4 how many times? And you would perform this

the same way you would do traditional long division. You’d say 2– you always start

with the highest degree term. 2 goes into the highest

degree term. You would ignore the 4. 2 goes into 2x how many times? Well, it goes into 2x

x times and you put the x in the x place. x times 2 is 2x. And just like traditional long

division, you now subtract. So 2x plus 4 minus 2x is what? It’s 4, right? And then 2 goes into

4 how many times? It goes into it two times,

a positive two times. Put that in the constants

place. 2 times 2 is 4. You subtract, remainder 0. So this might seem overkill

for what was probably a problem that you already

knew how to do and do it in a few steps. We’re now going to see

that this is a very generalizable process. You can do this really for any

degree polynomial dividing into any other degree

polynomial. Let me show you what

I’m talking about. So let’s say we wanted to divide

x plus 1 into x squared plus 3x plus 6. So what do we do here? So you look at the highest

degree term here, which is an x, and you look at the highest

degree term here, which is an x squared. So you can ignore

everything else. And that really simplifies

the process. You say x goes into x squared

how many times? Well, x squared divided by x is

just x, right? x goes into x squared x times. You put it in the x place. This is the x place right

here or the x to the first power place. So x times x plus 1 is what? x times x is x squared. x times 1 is x, so it’s

x squared plus x. And just like we did over

here, we now subtract. And what do we get? x squared plus 3x plus 6 minus

x squared– let me be very careful– this is minus

x squared plus x. I want to make sure that

negative sign only– it applies to this whole thing. So x squared minus x squared,

those cancel out. 3x, this is going

to be a minus x. Let me put that sign there. So this is minus x squared minus

x, just to be clear. We’re subtracting

the whole thing. 3x minus x is 2x. And then you bring down the 6,

or 6 minus 0 is nothing. So 2x plus 6. Now, you look at the highest

degree term, an x and a 2x. How many times does

x go into 2x? It goes into it two times. 2 times x is 2x. 2 times 1 is 2. So we get 2 times x plus

1 is 2x plus 2. But we’re going to want to

subtract this from this up here, so we’re going

to subtract it. Instead of writing 2x plus 2,

we could just write negative 2x minus 2 and then add them. These guys cancel out. 6 minus 2 is 4. And how many times

does x go into 4? We could just say that’s zero

times, or we could say that 4 is the remainder. So if we wanted to rewrite x

squared plus 3x plus 6 over x plus 1– notice, this is the

same thing as x squared plus 3x plus 6 divided by x plus 1,

this thing divided by this, we can now say that this is

equal to x plus 2. it is equal to x plus 2 plus

the remainder divided by x plus 1 plus 4 over x plus 1. This right here and this right

here are equivalent. And if you wanted to check that,

if you wanted to go from this back to that, what you

could do is multiply this by x plus 1 over x plus 1

and it add the two. So this is the same

thing as x plus 2. And I’m just going to multiply

that times x plus 1 over x plus 1. That’s just multiplying

it by 1. And then to that, add

4 over x plus 1. I did that so I have the same

common denominator. And when you perform this

addition right here, when you multiply these two binomials

and then add the 4 up here, you should you get x squared

plus 3x plus 6. Let’s do another one of these. They’re kind of fun. So let’s say that we have– we

want to simplify x squared plus 5x plus 4 over x plus 4. So once again, we can do our

algebraic long division. We can divide x plus 4 into

x squared plus 5x plus 4. And once again, same

exact process. Look at the highest degree terms

in both of them. x goes into x squared how many times? It goes into it x times. Put it in the x place. This is our x place

right here. X times x is x squared. x times 4 is 4x. And then, of course, we’re

going to want to subtract these from there. So let me just put a negative

sign there. And then these cancel out. 5x minus 4x is x. 4 minus 0 is plus 4. x plus 4, and then you could

even see this coming. You could say x plus 4 goes into

x plus 4 obviously one time, or if you were not looking

at the constant terms, you would completely just

say, well, x goes into x how many times? Well, one time. Plus 1. 1 times x is x. 1 times 4 is 4. We’re going to subtract them

from up here, so it cancels out, so we have no remainder. So this right here simplifies

to– this is equal to x plus 1. And there’s other ways you

could have done this. We could have tried to factor

this numerator. x squared plus 5x plus

4 over x plus 4. This is the same

thing as what? We could have factored this

numerator as x plus 4 times x plus 1. 4 times 1 is 4. 4 plus 1 is 5, all of

that over x plus 4. That cancels out and you’re

left just with x plus 1. Either way would have worked,

but the algebraic long division will always work, even

if you can’t cancel out factors like that, even if

you did have a remainder. In this situation, you didn’t. So this was equal to x plus 1. Let’s do another one of these

just to make sure that you really– because this is

actually a very, very useful skill to have in your toolkit. So let’s say we have

x squared– let me just change it up. Let’s say we had 2x squared–

I could really make these numbers up on the fly. 2x squared minus 20x plus 12

divided by– actually, let’s make it really interesting, just

to show you that it’ll always work. I want to go above quadratic. So let’s say we have 3x to the

third minus 2x squared plus 7x minus 4, and we want to divide

that by x squared plus 1. I just made this up. But we can just do the algebraic

long division to figure out what this is going

to be or what this simplified will be. x squared plus 1 divided into

this thing up here, 3x to the third minus 2x squared

plus 7x minus 4. Once again, look at the

highest degree term. x squared goes into 3x to the

third how many times? Well, it’s going to go

into it 3x times. You multiply 3x times this,

you get 3x to the third. So it’s going to go

into it 3x times. So you have to write the 3x over

here in the x terms. So it’s going to go into it 3x

times, just like that. Now let’s multiply. 3x times x squared is 3x

to the third, right? 3x times squared plus

3x times 1. So we have a 3x over here. I’m making sure to put

it in the x place. And we’re going to want

to subtract them. And what do we have? What do we have when

we do that? These cancel out. We have a minus 2x squared. And then 7x minus– because I

just subtracted 0 from there– 7x minus 3x is plus 4x,

and we have a minus 4. Once again, look at the

highest degree term. x squared and a negative

2x squared. So x squared goes

into negative 2x squared negative 2 times. Negative 2, put it in

the constants place. Negative 2 times x squared

is negative 2x squared. Negative 2 times 1

is negative 2. Now, we’re going to want to

subtract these from there, so let’s multiply them

by negative 1, or those become a positive. These two guys cancel out. 4x minus 0 is– let me switch

colors– 4x minus 0 is 4x. Negative 4 minus negative 2 or

negative 4 plus 2 is equal to negative 2. And then x squared, now it has

a higher degree than 4x, and the highest degree here, so we

view this as the remainder. So this expression we could

rewrite it as being equal to 3x minus 2– that’s the 3x minus

2– plus our remainder 4x minus 2, all of that

over x squared plus 1. Hopefully, you found that

as fun as I did.

It is excellent 🙋 and very understandable

😄

Can someone explain why, in the check for the first one, he multiplies (x+2) by (x+1)/(x+1)+4/(x+1)? Why did he add the x+1/x+1? Why not just (x+2)(x+1)+4/x+1??

I recommend trying to solve the equations yourself after the first one if you can't get it right check what you forgot to do by continuing to watch the video. It helped me and I got the method in the first watch! 😁

what if the coefficient in the denominator is greater

Thanks Mr.Khan!

I dont understand why we can ignore the other terms and only focus on the highest degree term

This was in my algebra 1 book. Is it algebra 1 or 2?

It give me extra knowledge and more confident to face any long division problem. If you don't believe me, ask me any questions from Division Algorithm for polynomial. I'm ready

Do we even need to learn algebra?😣😣😭😭😭😭😦

Great video!

Hi ! I'm looking for the software used to make this video if someone know the name

2 goes into 2x x times…. what?

Still don't get it. This way is harder than the way my professor teaches it D:

there's a shorter way of doing it, why even bother learning the longer AND more annoying way?

i'm still questioning when we use this irl

I have 12n2+28n+15 divided by 12 help please

That was fun to you? It was painful 😖

Ight lets see how this quiz goes tomorrow….*gets a 60

so if you divide x squared by x, you don't carry over the exponent? same with 2x divided by x? no more variable?

Seriously you should replace my Math teacher.

2×2=4-1=3 wiiik mafs

X+5/-26=4? pls simplify..

Just curious… Where would we see this or use this on real life?

Pray tell- how does 2 go into 2x "X times"??

That seems to be a remainder rather than the factorization.

they are kinda fun-Khan academy 2010

First

Wow you are pretty much better than any teacher!!

I don’t know how to do this ahaha FAKE LAUGHS then actually cries 😭😭😭😭

Thank you so much 🙂

Thank you SO MUCH for explaining this is a clear and concise manner.

Fun…..

I’m never ever going to study Math. No matter how hard I try, I’m still super clueless! I guess me and Math were never supposed to be fit for each other xD

i didnt understand! have a math test tommorwwwwwwww!

criesGod bless you

Thank you very much Khan Academy! My Math exam is on Wednesday and this Video has really helped me out A LOT! ThAnks So mUCH!

dang u got good mouse skills😍

I hate math. The only reason why I am doing this is because my education is important to me.

Does it worry anyone to have to perform this on a test. It does me.

thank goodness for videos like this. I forgot about this topic and this is my go-to channel from now on.

x10 y5 divide x4 y2=???

In the last week example, when you divide the higher degree polynomial by the quadratic equation, why, when you multiply the 1 in, (x+1), with the 2 in, (3x-2), it the two a positive and not negative as illustrated.

Guys right a variable cant be in a denominator? Why is it valid in this equation?

Some people would get really confused, please say that every subtracting time,the must subtracted numbers' signs need to be changed with it's opposite signs, same like you're subtracting a polynomial, the must subtracted numbers' sign must be changed to it's opposite sign.

THANK YOUUUUUU

Hi Mr. Khan! I just want to tell you that I got a 100% for my test. I was missing out in class by a lot and I watched your videos and got perfect for my second quiz of the year. I got a 10% on the first test, but after watching your videos I got a perfect score on the second one! THANK YOU SO MUCH… YOU'RE THE BEST!!!!!!!!!!!!!!!!!

Isn't this the guy who explained what the Federal funds rate is?

thank you

12 minute well spent thanks.

Thank youuu. 💜

ahhh… the first khan academy video i have ever watched

:(( I still dont get the long division thing

Geez I guess Im gonna fail my quiz again

Your 12 min video explains this more clearer than my teacher talking about it for 30 min

This is very simple, but the way you write it confuses me.

What if the 3rd example was divided the opposite way…

for a student who has a teacher who seemelesly breezes through the material, this is a

life-saver.Thank you!

This is hurting my brain

This method worked very well!

Mind-blown. It really helped that you showed it from the very beginning in simpler terms and then show what you found after by putting it back into the original, especially cos i was commmppllleetteelllyyy unaware of that division method until today…i just put the remainders next to the digit (which i tried to apply to this and failed miserably) xD

I spent a good portion of today in confusion and mild despair at the first bloody chapter!, feel a bit better now that i've understood this v_v'

Better than teachers though teachers are lazy

You are a miracle

This guy makes math way easier.

Wow i learned more in this video than in 1 Year of math class

Wow. Fun.

“How many times does 2 go into 2x?” X!! That’s right… wat.

"lets do another one of these. They're kind of fun!" ~rewinds video to make sure I heard that right~

Has a divide of cancel terms.

They did surgery on a grape

"hopefully you found that as fun as I did"

still don't get it

Great help thanks a lot

OMG I watched this video many times before I understood. THANKS SOOO MUCH

It has been 15 years since I've done algebraic long division and this video helped me remember it. Thanks!

😱😵

I GET IT NOW THANKS SAL!!!

Very good khan academy you saved my final exams

This really helped me in my homework because i was having trouble with it and my teacher explained it weirdly

But what if you don't have to use long division?

My schoool just became a 1 to 1 school and now we use this alll the time

God bless Khan Academy.

This made no sense

His writing is so nice oml mine is just a mess

is this also known as division of algebraic expressions?

is this also known as division of algebraic expressions?

This won't help me pass the NJSLA

thanks sam, it was very helpful!!

The divide subtract multiply version is a lot easier than trying to find factors even when factors exist.

Just gets messy.

Let's do another one of these

It's kind of funI don't understand this at all

Nice

3:20 thank you for clarifying the minus thing, it made all the difference!

at 2.42 you look at the highest degree term, and its the "x": how do you know its the "x"? The value of the "x" could be less than 1, no ???

What If you’re doing division with non common variables like (u+du)/(v+dv)

When Sal Khan’s video can’t help you, no one can🥺😢😭.

My professor teaches things a different way rather than the vids I watch on YouTube. Waahhh😭

Took me 30 min to try and work this out with worked examples, no use. This video has been extremely helpful, thank you.

i would say the explaining is too fast. couldnt keep up even just writing it down. gon play it at 0.5 speed now lol

Why did you skip the -2xsquared and place the -3x underneath the +7x, in the last problem? I was taught differently.

Really helps

You have changed my life

The second problem was looking like the Irish flag.

They Said That Is 6 minus 0 is nothing the rational functions still making into a long division for algebra