long division polynomials

[stiff echo]

hey I need some help

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[stiff echo]

So I have the problem

x^3+3x^2-2x+5 divided by x+1

now I got the answer x^2+2x but it doesn't seem right

[pliant lagoon]

Well

The idea is

Well it's similar to dividing numbers

[stiff echo]

thats what i did

[pliant lagoon]

Wait

Do you know how number division works?

Not the process

The idea of what is it

[stiff echo]

Yea

[pliant lagoon]

If you have a/b with remainder

What is the result

[stiff echo]

What??

[pliant lagoon]

What is the part that's not the remainder

[stiff echo]

the quotient I assume?

[pliant lagoon]

That's its name, what is it though

[stiff echo]

The answer to a division problem

[pliant lagoon]

It's the biggest number c

So that a=bc+remainder

Where the remainder is between 0 and c

[stiff echo]

Dividend = divisor * quotent + remainder

[pliant lagoon]

Yep

[stiff echo]

I dont get it when you just say abcdefg

Confused me there for a second

[pliant lagoon]

Ah I see

Just names for numbers

The exact same idea goes for polynomials

[stiff echo]

I just haven't done long division in years

[pliant lagoon]

a,b,c,r

Can be polynomials

Only difference is what do "biggest" and "between" mean now

Right?

So, the biggest polynomial is the one with the highest degree

If you have two with the same degree, the one with a bigger coefficient

[stiff echo]

Yea you start with dividing X^3 by x

How many times does x go into x^3

[pliant lagoon]

And "between" means the degree is between 0 and the degree of b

[pliant lagoon]

Almost there

I'm explaining context for a reason

For why this works

Once you understand why it works, you can understand how to do it very easily

[stiff echo]

Yea you're giving me a hard time with this philosophy talk man

I just wanna know if I placed my answer in the right spots

[pliant lagoon]

It's mathematics, not philosophy

The goal isn't to simply verify your work

Because if we just do that, you'll forget polynomial division once you stop using it

[stiff echo]

I just have an evaluation in a few days and I am studying for it

[pliant lagoon]

The goal is to understand what are you doing, so that you can verify your own work easily

If you don't want me to help with that you should wait for someone else to check your numbers, I don't believe it will help with anything

@stiff echo are you willing to follow through with this?

[stiff echo]

Yea sorry I had to go take a #2

So I dont get this entire "between 0 and c"

[pliant lagoon]

Oh, well

We could pick any number

As the divisor

And just choose the fitting remainder so it holds

But we want the remainder to be small

[stiff echo]

So let's pick the nimber 90

number*

the divisor is 9

[pliant lagoon]

How small can we guarantee? No bigger than c-1

[stiff echo]

wait no let's have number 91, divisor 9

[pliant lagoon]

Aight

Lets have 98 actually

You'll see why

So,

98=9c+r

Find a number c

So that r is the smallest positive number you can find

[stiff echo]

Wait c is the quotient right?

[pliant lagoon]

Yep

[stiff echo]

well in this case it's obviously 10 quotient and the smallest number will be 8 (r)

[pliant lagoon]

Right

[stiff echo]

It can't be anythign under 8 since it wouldn't make sense

[pliant lagoon]

I claim

[stiff echo]

anything*

[pliant lagoon]

That no matter what number

The remainder is always between 0 and 8

If it goes over 8, it's not the smallest

If it goes below 0, it's not ≥0

Right?

[stiff echo]

Wait how is it between 0 and 8 if the only valid remainder is 8?

It cant be 7 since that wouldnt equal 98

[pliant lagoon]

Well for 98 it's 8

But for 99 it's 0

For 97 it's 7

However there is no number so that r=11 for example

[stiff echo]

yea that goes past 8

[pliant lagoon]

So, same with polynomials

[stiff echo]

I tested my answer with the division statement and it didnt give the right answer, god damn it

[pliant lagoon]

The difference is instead of size, we use degree

[stiff echo]

Wdym size? like the size of the remainder?

[pliant lagoon]

p(x)=g(x)q(x)+r(x)

[pliant lagoon]

Yeah

[stiff echo]

So instead of the remainder we use degree alright

So in the problem I posted the highest degree is 3

[pliant lagoon]

Well the size of the remainder is the degree

[stiff echo]

OH WAIT so the highest remainder is the highest degree?

So if it's 3 the highest remainder it can be is 3??

[pliant lagoon]

Wdym

[stiff echo]

If the highest degree (x^3) in this case is 3 the highest remainder it can be is between 0 and 3?

[pliant lagoon]

Well

0 and 3-1

It always goes one below

Like with number division

It's not 9, it's 8

Same with degrees

It's between 0 and deg(g(x))-1

So if we divide $x^{400}+x$ by $x^2+1$, the remainder is at most of degree $2-1=1$

[little musk]

yoavmal

[pliant lagoon]

Do I make sense?

[stiff echo]

Let me re-read a few times

So the highest remainder it can be is 1

[pliant lagoon]

Yep

Now, lets try find an algorithmic approach

To finding the quotient

We can try again with number division first

What we can do is start by having

[stiff echo]

So the long division with polynomials is like an algorithm?

[pliant lagoon]

a=0c+a

[pliant lagoon]

Exactly!

[pliant lagoon]

Now, we take some part of a

And transfer it to c

For example with our system from earlier

98=0*9+98

[stiff echo]

If the divisior is 0 then doesn't it all just become 0?

or is it undefined

[pliant lagoon]

This isn't the final division form

Since r isn't between 0 and 8

So, lets ask ourselves something to make our life easy

What is the biggest power of 10

So that, that power * 9

Is less than 98?

[stiff echo]

Wait so R here is 98?

[pliant lagoon]

For now

It's not the final form

[stiff echo]

Wdym by biggest power of 10

Where did 10 come from

[pliant lagoon]

We work in base 10

So it makes our life easy to divide using it

98 is actually 9*10+8*1

[stiff echo]

What is base 10?

[pliant lagoon]

827 is actually 8*100+2*10+7*1

[pliant lagoon]

The system we use for representing our numbers

[stiff echo]

Never heard of it before

[pliant lagoon]

There are different bases you can use, which give different values to different sequences of digits

[pliant lagoon]

Does this make sense to you though

[stiff echo]

Not really

[pliant lagoon]

Hmm

[stiff echo]

Just kinda side tracked from the whole between 0 and r

[pliant lagoon]

Then you have a different issue

Nevermind, not entirely relevant

Just, take it that we use 10 for that

We have different powers of 10

1, 10, 100, 1000

Etc

[stiff echo]

Yea

[pliant lagoon]

The question is

What is the biggest power of 10 we can take

[stiff echo]

For 98?

[pliant lagoon]

So that 9 times that power of 10

[stiff echo]

The highest is 10

[pliant lagoon]

Is less than 98

[pliant lagoon]

Yep

So we can say

98=9*10+(98-9*10)

=9*10+8

By this point we know we can stop because 0≤8<9

[stiff echo]

So how does that relate to the initial problem I had

[pliant lagoon]

But if it were 827 it would be different

[pliant lagoon]

Well, instead of 10

[stiff echo]

I dont see any use of power of 10s stuff

[pliant lagoon]

We just use powers of x

[stiff echo]

the x from x+1?

[pliant lagoon]

Well not from x+1 in specific

Just x in general

So lets try do the division the same way

[stiff echo]

Okay so how many times does x^3 go into x

it becomes x^2

[pliant lagoon]

We want to get

[stiff echo]

because x^2 * x is x^3

[pliant lagoon]

Wait what were the polynomials

()/(x+1)

What was the left one?

[pliant lagoon]

x³+3x²-2x+5?

[stiff echo]

Yea

[pliant lagoon]

Ok, so we want

x³+3x²-2x+5=(x+1)q(x)+r(x)

[stiff echo]

Yea

[pliant lagoon]

Right?

Lets do exactly the same, and you'll see why it's actually what you were doing the whole time

And also see how it helps you spot your mistake very fast

So we start with x³+3x²-2x+5=(x+1)*0+x³+3x²-2x+5

[stiff echo]

Uh what?

[pliant lagoon]

Here we use r=x³+3x²-2x+5=(x+1)*0+x³+3x²-2x+5

[stiff echo]

That isn't long division

[pliant lagoon]

It's not the final form

[pliant lagoon]

It is, you'll see

It's exactly long division

[stiff echo]

It doesn't have the little thing that looks like the square root symbol

[pliant lagoon]

The symbols may be different, the process is identical

[stiff echo]

You're confusing me with the whole r = the entire polynomial

[pliant lagoon]

We start with r=the entire thing

And slowly subtract from it

[stiff echo]

What do we subtract

[pliant lagoon]

When it was numbers

We had the highest power of 10, so that that times 9 was less than 98

Here

We have the highest power of x

So that that times (x+1) is less than x³+3x²-2x+5

Do you see it

Or is it a bit confusing still?

[stiff echo]

The highest power is x^3

[pliant lagoon]

Well, lets try

(x+1)x³=x⁴+x³

That has a degree of 4

So it's more than x³+3x²-2x+5

[stiff echo]

That isn't long division bruh

[pliant lagoon]

It is

[stiff echo]

long division is when you move it down and stuff

[pliant lagoon]

It's just a different way to write it

We didn't do the moving down

Yet

The highest power would be x²

[stiff echo]

Like this^^^

Thats how you do it

[pliant lagoon]

So (x+1)x²

[pliant lagoon]

=x³+x²

Does it look familiar now?

[stiff echo]

Where did x^2 come from

[pliant lagoon]

Well, x³ was too big

[stiff echo]

Are you talking about the orange one??

Like in my screenshot?

[pliant lagoon]

It would be the same orange one

[stiff echo]

Yes x only fix x^2 into x^3

yes I get that now

We then multiply that same one by (x+1)

x^3 and then x^2

[pliant lagoon]

Ok, so now we write the whole thing again

[stiff echo]

the x^3 cancels out

becomes 0

[pliant lagoon]

x³+3x²-2x+5=(x+1)*0+x³+3x²-2x+5

[stiff echo]

3x^2 minus the x^2

[pliant lagoon]

Becomes

[stiff echo]

We get 2x^2

We are now stuck with that what I showed above

Am i doing it correctly?

[pliant lagoon]

x³+3x²-2x+5=(x+1)*x²+(x³+3x²-2x+5-(x+1)*x²)

[stiff echo]

Dude like I cant read it like that, I can't form that in my head like I did above

[pliant lagoon]

I see hmm

[stiff echo]

You'd need to write it how I do it in order for me to understand

Like you get what I am saying?

am I doing it correctly above?

[pliant lagoon]

Yes I do

[stiff echo]

I am doing everything you said before

[pliant lagoon]

What you are doing always corresponds to an equation like we have

With p(x)=g(x)q(x)+r(x)

q(x) is the thing above the line

[stiff echo]

g(x) is the divisior

that's left of the dividend

[pliant lagoon]

Yep

Exactly

And the thing you keep taking down

[stiff echo]

the quotient is on top

[pliant lagoon]

That's r(x)

You start with r=p

But keep subtracting from it

[stiff echo]

Wait wait so let me re-read this again

So the first part

is the dividend

[pliant lagoon]

Yes

[stiff echo]

equals the divisor

Which is x+1

and you did the multiply by x^2 I get that

Now the remainder is what's under

that last paren

with the 2 plynomials

[pliant lagoon]

Correct

[stiff echo]

Now I get it

[pliant lagoon]

Was it worth the effort?

[stiff echo]

Yea I get how to read that part now

It's just a lot of exponents and shit

[pliant lagoon]

So, basically every step

[stiff echo]

Okay now next since x^3 cancels out

that becomes 0

[pliant lagoon]

After the subtraction

[stiff echo]

the 3x^2 minus 2x^2 is x^2 and we bring down the 2x

[pliant lagoon]

You get an equation you can verify

So first equation is $x^3+3x^2-2x+5=(x+1)\cdot0+(x^3+3x^2-2x+5)$

Second equation is

[stiff echo]

Do I do this now??

[little musk]

yoavmal

[pliant lagoon]

Second equation is

[stiff echo]

Where did the 0 come from

[stiff echo]

x+1 just becomes 0 what

[pliant lagoon]

Before you wrote anything above the line

It was 0

$x^3+3x^2-2x+5=(x+1)\cdot x^2+(2x^2-2x+5)$

[little musk]

yoavmal

[stiff echo]

Oh yeah

[pliant lagoon]

That is the second equation

Now what you're doing is just secretly dividing 2x²-2x+5 by x+1

[stiff echo]

this is where I put the first thing on top of the line yea?

[pliant lagoon]

The same way we just did

[pliant lagoon]

Yeah the x²

[stiff echo]

Quick question when doing the long division, do you recommend bringing down all the terms down?

or only one of them

Or just all of them to form one polynomial

[pliant lagoon]

Whatever you find more convenient

[stiff echo]

Like I brought -2x here but left 5

Yea I'll just bring them all down then

So now we divide that new polynomial by x+1

and the same thing repeats

[pliant lagoon]

It's really irrelevant, all that matters is you don't make mistakes or work too hard

[pliant lagoon]

Exactly, what do we get then?

[stiff echo]

Wait what if the term goes outside the line

[pliant lagoon]

$2x^2-2x+5=(x+1)\cdot0+(2x^2-2x+5)$

[little musk]

yoavmal

[pliant lagoon]

Wdym

[stiff echo]

Like where does the blue go

I brought it down

[pliant lagoon]

Oh, well it can't

You always bring one term in

[stiff echo]

and just to make sure when dividing we only focus on the X and ignore the 1?

[pliant lagoon]

Not all three

It should cancel out by the end

[stiff echo]

Oh alright

[pliant lagoon]

If it doesn't, you have a remainder

And you just write it as that

Ok so

[pliant lagoon]

Lets try do that

[stiff echo]

Yea one sec

How do I know which term the quotient starts to go on top of

If I do 2x^2 divided by the x do I put the answer on top of the 2x^2 or the -2x?

[pliant lagoon]

(It doesn't matter)

[stiff echo]

Like this^^?

Is that the answer?

[pliant lagoon]

Yeah

[stiff echo]

After it's just remainder 5

[pliant lagoon]

Not yet

Lets verify the equation

[pliant lagoon]

Did you do this?

[stiff echo]

Yea I showed a screenshot

[stiff echo]

Here

I did it seperatly

It's just 2x with remainder 5

[pliant lagoon]

Oh I just realized

[stiff echo]

[pliant lagoon]

Lets verify

(x+1)(2x)+5

[stiff echo]

So quotient times divisor plus remainder

[pliant lagoon]

=2x²+2x+5

[stiff echo]

Wait no it's not 2x it's x^2+2x

[pliant lagoon]

Is that 2x²-2x+5

[stiff echo]

x^2+2x

is the quotient to the original problem

[pliant lagoon]

I'm checking for the smaller one right now

Since that's your error

[stiff echo]

Oh it works!

[pliant lagoon]

It doesn't lol

[stiff echo]

Wait what

[pliant lagoon]

Is that

[stiff echo]

Bro the fucking minus bruh

[pliant lagoon]

And that

[stiff echo]

How do I make it fucking minus

[pliant lagoon]

You don't

[stiff echo]

So leave it like that?

[pliant lagoon]

The difference is just not 5

[stiff echo]

There's nothing we can do

[pliant lagoon]

There is

What you said is

(2x²-2x+5)-(2x²+2x)=5

It's -4x+5

[stiff echo]

Where did you get 2x^2 from

What equation is that

How did you get to that

[pliant lagoon]

Here

Black section

[stiff echo]

the bottom one?

[pliant lagoon]

Yeah

[stiff echo]

Oh yeah I see now

So that is the divisor * quotient

move to the other side with the dividend

remainder stays on right side of equals sign

[pliant lagoon]

Yeah that's how we calculate the remainder each time

[stiff echo]

Wait so what even is this then

[pliant lagoon]

So

[pliant lagoon]

Remainder

[stiff echo]

Let me check myself

-4x+5 = 5

Then what?

[pliant lagoon]

Not =5

[stiff echo]

what why? it said = 5

[pliant lagoon]

The remainder is just -4x+5

[stiff echo]

Oh

[pliant lagoon]

You said that

[stiff echo]

wait wait so x^3+3x^2-2x+5 divided by x+1 is x^2+2x remainder -4x+5?

[pliant lagoon]

That is a way to say it but

The degree of x+1 is 1

The degree of -4x+5 is 1

So the division is not yet done

[stiff echo]

Bruh are you shitting me there's more division?

This is the 1st question to my homework bruh

[pliant lagoon]

Yeah

[pliant lagoon]

It feels longer because we're making a lot of explanations and checks for each step

In practice it's pretty short

[stiff echo]

Isn't the remainder theorem able to just find the remainder

f(-1) there

[pliant lagoon]

It is but

We want the quotient

[stiff echo]

if we have the dividend, divisor and remainder can't we find the quotient with all three without needing to even do the long division stuff?

Galaxy brain moment?

[pliant lagoon]

You're mostly correct but

This only works for division of the form x-r

Dividing by x²+bx+x no longer works

[stiff echo]

Yea if the leading coef is greater than 1 then we're cooked

Or wait no so i thas to be a binomial?

it has*

(x-p)

[pliant lagoon]

Idk I haven't slept in nearly 24 hours

[stiff echo]

This discord thread has to be one of the biggest learning curves I've had in my 3+ years of highschool.

[stiff echo]

What grade are you in

[pliant lagoon]

I'm in 2nd year in university

Finishing

[stiff echo]

Bro how tf do you know this shit so well

99% of the time everyone in my school learns this, does the test and forgets right after

How do you make sure to remember this stuff for longer periods of time

Wait btw bro I need some help with another problem

It's in the same question I just dont know what it's asking

[pliant lagoon]

What do you think are we doing right now

You won't remember the technique

But the concept of

"Fitting the biggest polynomial"

[pliant lagoon]

That does stick in the bsain

[stiff echo]

I see

[stiff echo]

What does it mean by restrictions

[pliant lagoon]

No idea

Maybe illegal values for x?

[stiff echo]

Oh yea

[pliant lagoon]

Like, can't divide by 0

[stiff echo]

Wait how would it look for the quotient anyway

x^2+2x

move everything to the other side but x

x = √(-2x)

≠*

Is that correct?

[pliant lagoon]

Well you're not done remember

[stiff echo]

So we're not even done the division?

[pliant lagoon]

No, just one step more

[stiff echo]

Whats the last step

[pliant lagoon]

Dividing -4x+5 by x+1

[stiff echo]

Ohhh yeah

wait but -4x+5 is the remainder why would we divide it

[pliant lagoon]

Since we want the degree of the remainder to be less than the degree of the divisor

Less than 1

[stiff echo]

Is that good?

That should be correct

[pliant lagoon]

Does

(-4x+5)-(-4x-4)=1?

[stiff echo]

shit wait no

the remainder is 9

There we go

Finally

9 remainder

https://tenor.com/view/cristiano-ucl-kiss-cr7-cristiano-ronaldo-goat-ucl-gif-14851486156773034728

[pliant lagoon]

Lets verify

(-1)³+3(-1)²-2(-1)+5

=-1+3+2+5

=9

[stiff echo]

Let's go dude

[pliant lagoon]

So what's the final equation?

$x^3+3x^2-2x+5=(x+1)(x^2+2x-4)+9$

[little musk]

yoavmal

[pliant lagoon]

?

[stiff echo]

One sec im sending

Like that?

Bro how did you get -4 in there

Like HOW

[pliant lagoon]

.

[stiff echo]

Oh shit yea

[pliant lagoon]

Yeah just gotta be attentive

[stiff echo]

Dude this is a crazy learning curve for me

[pliant lagoon]

I'm glad it is

[stiff echo]

It's just our teacher told us to do this and get remainder

[pliant lagoon]

When we finish i'm dividing sobbed though

[stiff echo]

Wdym

[pliant lagoon]

I haven't slept in like

24 hours

I'm really tired so when we finish i'll go sleep

So want to retry doing the division, but this time without side steps?

[stiff echo]

Can we try it with the next problem?

[pliant lagoon]

So you see it's not that slow

Sure

[stiff echo]

Let me try to write the equation

[pliant lagoon]

Aight

[stiff echo]

3x^4-4x^3-6x^2+17x-8 = (3x-4) Q(x) + R(x)

So like this

The first one will have the quotient 0 obviously

and remainder is also 0?

[pliant lagoon]

Just add brackets

(3x-4)

[stiff echo]

Alr alr

The next one switches from 0 to x^3 so thats the 2nd step

[pliant lagoon]

No, equation has to be true

Remainder must be the whole polynomial

At the first step

[stiff echo]

So (3x-4) Q(x) + the whole polynomial (r)?

Oh wait I remember why

because thats all we have under the line thingy

and it changes throughout the solving process

The next one is lemme type it out

3x^4-4x^3-6x^2+17x-8 = (3x-4) (x^3) + 3x^4-4x^3

that one should be good right?

[pliant lagoon]

Seems off

[stiff echo]

Which part specifically?

[pliant lagoon]

Left side has 3x⁴

Right side has 6x⁴

[stiff echo]

Left and right you mean that?

[pliant lagoon]

Left and right of this equation

[stiff echo]

Ohh alr

[pliant lagoon]

That seems right though

[stiff echo]

How does 6x^4 appear?

[pliant lagoon]

Here?

3x*x³

+3x⁴

On right

Sorry i'm falling asleep

[stiff echo]

OH I see

I see it

[pliant lagoon]

I must leave before I get too tired

Alright?

[stiff echo]

yea that's fine

i can solve it on my own

[pliant lagoon]

Thanks, sorry but it's like

5:36

[stiff echo]

Yea all good man

[pliant lagoon]

am

[stiff echo]

Can other people help though?

[pliant lagoon]

If they see this, yes

[stiff echo]

@runic sonnet Let me post it here

Where did the 30/x+2 come from

[runic sonnet]

It’s the remainder

Then u have to divide it anyway

[stiff echo]

Bro I got remainder 4

Divide the remainder by divisor correct?

[runic sonnet]

Yea

[stiff echo]

Yea so it should be 4 anyway

[runic sonnet]

Ok

Then what’s confusing u

If the answer is supposed to be 4

[stiff echo]

No no I just forgot you had to divide the remainder anyway

but I get it now

[runic sonnet]

And u know that u divide remainder by your divisor

[fallen glade]

wat

[visual olive]

is help still needed?

[pliant lagoon]

@stiff echo

[stiff echo]

Hi