help-0

[queen raven]

whoops, sorry i sent the thing at the exact same time

[boreal verge]

do ask your question

[woven bloom]

here

[queen raven]

flip the denominator and numerator for the divisor and times them together

then simplify

[woven bloom]

i was stuck on the simplify bit tho

[queen raven]

factorise

[boreal verge]

i do hope you know your formulas for factoring

like diff of squares etc

[woven bloom]

i do

lemme send what i was up to

oh wait hold up

i got it

thanks guys

.close

[lone heart]

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[queen raven]

lol

i need help with this

[boreal verge]

wait a bit so the channel resets then send a message

[queen raven]

can you help me out in help 18?

[lone heart]

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[alpine sable]

Hi

[alpine sable]

A truck tire has a diameter of 22 inches if the truck completes half a turn how far would the truck have moved

I need help

[wild trail]

!status

[lone heart]

What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[wild trail]

@alpine sable

[alpine sable]

4

[wild trail]

Show your work then

[alpine sable]

I used c = pi x 22 I got 69 cm

Circumference equation

[wild trail]

alright, then

[wind cloak]

nice

[alpine sable]

Is that correct ?

[wild trail]

Not quite but you're almost there

Remember, the truck completes "half" turn

[alpine sable]

69/2

[wild trail]

yep

[wild trail]

Btw what value of pi did you take?

[alpine sable]

I got 22pi

Changed it

[wild trail]

So, you took pi as 3.14 and then rounded the answer?

[alpine sable]

No

I just did it on a calculator

It had the pi option

[wind cloak]

,calc pi * 22

[ocean seal]

Result:

69.115038378975

[wind cloak]

Seems legit

[wild trail]

Kinda weird that it gave a whole number not decimal

No worries, I was just curious

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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@steep widget Has your question been resolved?

[steep widget]

<@&286206848099549185>

[languid canopy]

apply the formula of tan inverse a - tan inverse b

[steep widget]

They weren't the one who tagged

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[radiant dirge]

is this true $\cup \emptyset = \cup {\emptyset}$

[ocean seal]

ta

[pliant cedar]

union of the empty set with what?

[radiant dirge]

its supposed to be this but idk the latex code for this

[delicate jacinth]

that is union of all elements of M

[radiant dirge]

yeah

so replace the cup with this symbol

[delicate jacinth]

The left side is the union of all elements of a empty set

right side is the empty side

[radiant dirge]

so its $\bigcup \emptyset = \emptyset = \bigcup {\emptyset}$

crap

[wind cloak]

$\bigcup$

[ocean seal]

NEONPerseus

ta

[delicate jacinth]

I think $\bigcup \emptyset$ doesn't exist

[ocean seal]

sopinha

[radiant dirge]

why so

if no element belongs to a set it is empty but it exists right

[boreal verge]

just as you cant say $A = \emptyset$, for any set A, that also implies the elemententary operations dont work when its treated as a set

[ocean seal]

mreow

[delicate jacinth]

I mean

intuitively it doesn't exist

because it would imply that the empty set has elements

but the axiom of the union states that it always exists

so yeah

https://en.wikipedia.org/wiki/Axiom_of_union

idk i'm asking my teacher tomorrow

[lone heart]

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[alpine sable]

hey

[alpine sable]

wait its uploading

i need help with the 1st one

idk how to find u1 and u2.

[slim swan]

ye your gonna have to translate that

[delicate jacinth]

,rotate

[alpine sable]

okay

wait

(Un) is a geometric sequence , fully increasing , her first step is Uo and her ratio is q

1 ) find u1 and u2 and then conclude the value of q

[alpine sable]

done

i translated that since we study in a completley diffrent language it took me a bit to confirm if i am using the right words

[late mango]

system of equations

with the info you're given about u1 and u2

[alpine sable]

ah right

i am going to try that

[alpine sable]

hey

i found the solution but i didnt understand it lol

[late mango]

so where are you specifically stuck at

[alpine sable]

the 1st question

[late mango]

ln(u1) + ln(u2)

do the specific formula then cancel ln from both sides

[alpine sable]

.close

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[frigid verge]

I understand logarithms but I dont really know what approach to take to this question

[oblique trail]

log(a) + log(b) = log(ab)

@frigid verge

[frigid verge]

Well

[minor needle]

#❓how-to-get-help

[frigid verge]

Do I just keep working on this?

[oblique trail]

No

There’s gotta be a better way

[frigid verge]

That's what I thought

[oblique trail]

,w sqrt(90+sqrt(90))

[ocean seal]

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[oblique trail]

Interesting

So sqrt(90) is actually 3sqrt10

Which you can simplify

[frigid verge]

I was gonna use that but it's the same kinda thing

[oblique trail]

But hm

let me try something quickly

,w (10+sqrt(10-x))(10+sqrt(10+x))

[ocean seal]

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[oblique trail]

Bruh what

[frigid verge]

😂

[oblique trail]

Idk

@frigid verge

So I plugged it into wolfram

[oblique trail]

I think they might just want you to do this

[frigid verge]

😂 that's hella mad

I'll see if anyone can see some stuff going on cuz I cant

I might just have to do it the long way

<@&286206848099549185>

^Same question as pinged^

[lone heart]

@frigid verge Has your question been resolved?

[frigid verge]

.close

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[solid bone]

hi there i tried finding an equivalence between the boxed boolean expressions in the picture attached. i tried manipulating it with boolean algebra but i couldn't prove they are equivalent. is there anything that i might be missing here?

[solid bone]

for reference i compared their truth tables and they were equivalent so i wasn't sure what i was doing wrong

mb for the weird naming scheme but yeah z corresponds to Cin in the written picture above

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[solid bone]

.close

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[dim matrix]

Shouldn't it be just Cos^2(theta)+Sin^2(theta) in the second last step?

[echo socket]

Seems to be a typo

[alpine sable]

yeah

[dim matrix]

Where should I go through trigonometry just for the sake of Complex numbers?

[lone heart]

@dim matrix Has your question been resolved?

[dim matrix]

<@&286206848099549185>

[dreamy sluice]

I don't get what you mean

[dim matrix]

I want to learn the concepts of Trign. just enough so that I can study Complex Numbers.

[dreamy sluice]

ohh

sorry

I can't help you with that, I guess

[dim matrix]

np, thanks. 🙂

[unique trout]

yeah?

[unique trout]

wait is this the question?

[dim matrix]

yeah

[unique trout]

oh

im not exactly sure

[unique trout]

possibly a basic understanding of trig and other stuff related to trig

i think

maybe also a good background of algebra

[dim matrix]

yes

[unique trout]

hope that helped!

[dim matrix]

thanks @unique trout @echo socket @alpine sable

.close

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[blazing glacier]

triangles

[blazing glacier]

triangles

i need some help with triangles

.close

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[gusty coral]

hi

[gusty coral]

question about mechanics, when are you allowed to use v=wr? and when are u not allowed to use the formula

[lone heart]

@gusty coral Has your question been resolved?

[rocky grove]

When you have angular velocity w and you want to convert it to normal linear velocity v

You use v = wr

And vice versa

[gusty coral]

but are there any conditions as to when you can use them? cuz i think one of the conditions was that there could be no slipping

[rocky grove]

Oh yea contact with the surface

And no energy loss

[gusty coral]

oh okay:) are there any conditions for conservation of momentum or angular momentum

[rocky grove]

Yep they need to be satisfied

[gusty coral]

what are the conditions for those formulas? i know for conserv of momentum there cant be any external forces but

are there more

[rocky grove]

I think the collisions should be fully elastic too

[gusty coral]

okay thenku:}

A ball dropped from a tall building at the equator falls vertically downwards. In which direction will the ball deviate during its fall due to the Coriolis force? does anyone know why its towards the east

[lone heart]

@gusty coral Has your question been resolved?

[gusty coral]

<@&286206848099549185> ☺️☺️

[oak pawn]

@muted rock

[gusty gorge]

no, that's conservation of energy, and all perfectly elastic means is that energy is not lost into other forms

conservation of momentum basically always holds if there's no external force on it

[gusty gorge]

are you sure? I feel like it would just drop straight down, albeit at a slower acceleration than g

[gusty coral]

Iguess im sure, i just saw it in the correction key idk:(

[gusty gorge]

well, you can say that the angular speed of the ball around the Earth's axis increases

that's because of conservation of angular momentum

but I'm not sure that actually makes the ball drop toward the east

I never understood rotating things lol

probably should check the physics server

actually yeah it does sorta make sense it would go toward the east

[gusty coral]

how exactly? i dont rlly understand :c

[gusty gorge]

basically if you consider just the ball going around the Earth's axis, the angular momentum has to be conserved because there are no torques on it

and when the ball gets closer to Earth's axis, the angular velocity has to increase to compensate for the reduced distance because the angular momentum is conserved

[gusty coral]

does coriolis have anyth to do with that?

[gusty gorge]

I don't think so

coriolis forces are north and south

but it's related

I'm talking from the frame of reference from someone just floating in space in an inertial reference frame

but on the Earth, you'd see this increase in angular velocity as a sort of fictitious force

[gusty coral]

Ahh i see, thanks:) if you dont mind, ive got a small q about this as well, its also about a rotational thing tho🙈

A gyroscope consists of a rotor mounted at the center of a thin axis. The gyroscope rotates with angular momentum as shown in the figure while the axis rests on a support that allows rotation of the axis. In which direction will the axis rotate around the support?

its a rlly dumb thing

but i just thought itd turn clockwise

w righthandrule

[gusty gorge]

not sure if I actually know what's spinning here lol

is the whole thing the gyroscope?

[gusty coral]

yes

i think the plate underneath

[gusty gorge]

doesn't that mean that the rotor is turning toward the reader?

[gusty coral]

thats what i thought too, but correction says , when you look frm above u see itd turn counterclockwise

[gusty gorge]

is the rotor slowing down or something

I honestly have no idea, would probably have to think for a while

you should ask the physics server in #old-network lol

[gusty coral]

ohh posted them there!

anyone know how we got R(1-cos(theta))?

(ik im asking sm q’s im so sorry,, last exam tom😭)

[gusty gorge]

hmm it comes from trigonometry

basically the resulting point is 90-theta degrees off the horizontal

and therefore it's sin(90-theta) degrees vertical distance off the horizontal

and that's cos theta

so the gravitational potential energy, relative to the halfway of the circle, is cos theta * R

at the top, the gravitational potential energy is R

so the diff is R - cos theta * R = R(1-cos theta)

[gusty coral]

i thought Rwas radius here

[gusty gorge]

it is

okay I was very sloppy here

Rmg is the gravitational potential energy at the top

R cos theta mg is the gravitational potential energy after you rotate by theta

if you subtract the two, the potential difference is Rmg(1- cos theta)

[gusty coral]

Omg thanku sm

If i mayyyy,, a q about dB🥲 i swear im almost done now

Two fireworks rockets exploding simultaneously produce a sound intensity of 95 dB at a certain location on the ground. What was the sound intensity if only one of the two rockets had exploded?

howd they get the 3dB

i think were using this formula but not sure how to get 3

[gusty gorge]

what is it, like 10 db is a 10-fold increase?

[gusty coral]

decibels

[gusty gorge]

yeah decibels

I forgot how they were defined lol

right so an increase in 10 dB corresponds to a 10x increase in intensity

so we have something like relative difference in intensity = 10^(db difference / 10)

taking log base 10 of both sides

you get db difference / 10 = log relative difference in intensity

db difference = 10 times log relative difference in intensity

for something that's twice as intense

you get

,w 10 * log base 10 of 2

[ocean seal]

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[gusty gorge]

or 3 dB

[gusty coral]

Thank u sososo much, really.

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[modern wyvern]

i dont understand how this is incorrect

[molten pivot]

i think it is best to give you an example of two functions

consider x^3

[modern wyvern]

kk

[molten pivot]

consider |x|

[modern wyvern]

k

[molten pivot]

does 3 hold for x^3?

[modern wyvern]

no

[molten pivot]

alright.

now does 5 hold for |x|?

[modern wyvern]

no

[molten pivot]

there you go

[modern wyvern]

so the last one is incorrect

[molten pivot]

all looks good actually, your answer for 5 was right, i misread

you should just have 1 and 2 checked in

[modern wyvern]

.closed

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[junior warren]

could somebody help me through this proof

[junior warren]

not sure how i did it wrong

[raven haven]

the left column

you missed one of them

[junior warren]

yeah but do either of them make sense to use

[raven haven]

of course

[junior warren]

but i am using contrapositive

both of those are for the original statement right?

[raven haven]

honestly I wouldn't use that

as in

personally I don't see the need for it but

apparently the online marking system does?

[junior warren]

put the assumption first maybe?

seems redundant but

[raven haven]

yeah I agree feels redundant

[junior warren]

because the therefore statement is just

not useful here at all

[raven haven]

but oh well

[junior warren]

i am doing nothing related to the same parity

i will try to throw the assumption first

hm

was also wrong

i mean the only other thing i could do is put the therefore after

but i think the rest of the proof is right?

weirdly enough that was it

thanks

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[radiant lantern]

I need help with a physic question

[radiant lantern]

A runner can reach the maximum speed of 12.8m/s. Starting from rest, it accelerates at the rate of 3.20m/s2.

a) How long does it take to reach its maximum speed?

b) If he runs 200m, how far does he have to go when he reaches his maximum speed?

c) If we consider that from this moment, this runner decelerates at the rate of 1.00m/s every 5.00s, after how long after his start will he cross the finish line?

[median oar]

What formulas do you know regarding this topic?

[radiant lantern]

these ones

[median oar]

Part a

What info do you have

And what are you trying to find

And which formula matches up for that

[radiant lantern]

this one and i'm trying to find t

I understand the a) and the b) but I don't understand how to do the c)

[median oar]

Ok part c

Let’s split this into 2 parts

When does he hit his top speed

(Both time and distance)

(So what have you got as the answers to part a and b)

[radiant lantern]

a gives 4s and b gives 174,4m

[median oar]

Ok how much distance does he have left

[radiant lantern]

174.4m I think

[median oar]

Is that how much distance is left or how much distance has been covered

[radiant lantern]

it's the distance left

[median oar]

Ok

So this is the 2nd part

First part is accelerating

That took 4s

He’s now at 12.8m/s

He’s now decelerating at 1m/s per 5s

That means we have initial velocity

Acceleration

And distance

Which formula uses those 3 to solve for time?

[radiant lantern]

this one

[median oar]

Ok

Plug in your numbers

And solve for t!

[radiant lantern]

but what is a ? it's -1,00m/s ?

[median oar]

$\frac{\frac{1 meter}{1 second}}{5 seconds}$

[ocean seal]

Frosst

[median oar]

It’s just fractions

Simplify it

[radiant lantern]

oohh so -0,2m/s ?

[median oar]

Nope

It’s an acceleration

Your units is for velocity

[radiant lantern]

it's -0,2m/s*2 then ?

[median oar]

Not *

^ is for power

[radiant lantern]

oh ok

[median oar]

• is multiply

[radiant lantern]

thank you I understand now

[median oar]

👍

[radiant lantern]

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[old mirage]

(I typed this in #proofs-and-logic as well, but this might be the more appropriate place for it)

[old mirage]

So my initial prompt says if x is odd then 2 does not divide x
(Such that k is within the set z)
It wants me to prove the contrapositive and to my new knowledge, that's if 2 does divide x then x is even
Can I use that "then x is even" in my proof? As in use the definition of evens in my proof to prove the initial prompt?

[placid zinc]

Not clear what your question is. What's a prompt?

You can use anything in a proof, as long as you're communicating clearly

[molten pivot]

what's the definition of even

i feel like this is more of a linguistic question than a mathematical one

[raven haven]

Please post the original question, exactly as it is.

[molten pivot]

unless of course it's just an exercise in logical implications then ok

[raven haven]

if n is even we have k in Z st n=2k

[molten pivot]

yeah exactly

so you're basically trying to prove that if 2 divides x then 2 divides x

like i said this is linguistics not math really

[raven haven]

[rose raven]

if 2 divides x then x is not odd

is the contrapositive

[raven haven]

yeah I think the OP got that

[rose raven]

ah i see

maybe i need the lesson in linguistics

[raven haven]

not actually linguistics

don't worry

[old mirage]

Yea I know I need to do something with n=2k

but like I'm just having a hard time piecing it together with the question ig

[raven haven]

,rccw

ah whoops

,rcw

[old mirage]

My bad thx lol

But I’m curious to know if I can use all the info that the contrapositive gives me

[raven haven]

Yes

[old mirage]

Cool

[raven haven]

You should use the hypothesis

If P then Q means you assume P and show Q

so you may assume 2 divides x

And since 2 divides x you can write it how you want, yeah

x=2k

[old mirage]

Gotcha

Thank u

I’m sorry if my question was confusing lol I just didn’t know what info I could use and what I couldn’t

.close

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[formal bone]

Find all solutions to the equation in the internal [0, 2π)

cos4x-cos2x=0

[thorny root]

"quadruple angle formula" and double angle formula

the first one is just double angle formula twice

[formal bone]

👍 thank you

[lone heart]

@formal bone Has your question been resolved?

[lone heart]

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[burnt escarp]

find the general term of the sequence 1,1000,10000000,1000000000000000

[burnt escarp]

I had that question on my math test and I couldn't figure it out can anyone help me?

[thorny root]

1 × 10⁰, 1 × 10³, 1 × 10⁷, 1 × 10¹⁵ I think

hmm

if the first term was 10 I'd say $a_{n+1} = 10(a_n)²$

[ocean seal]

biggboy

[thorny root]

but no clue on this one

[lone heart]

@burnt escarp Has your question been resolved?

[alpine sable]

Well we can think of it has
0,3,7,15

[thorny root]

oh I just noticed the pattern

go on tho

[alpine sable]

Well I'm thinking ahah

[thorny root]

one less than powers of two :P

[alpine sable]

Na

[thorny root]

no?

[alpine sable]

Not working for the first

[thorny root]

0 is 1 - 1

[alpine sable]

But for the other its good

[thorny root]

2⁰ - 1

[alpine sable]

Yes and 2^1-1=1

[thorny root]

yeah

[alpine sable]

Not equal 3

[thorny root]

oh

we skip one I see

hmm

[alpine sable]

But I did think of it first

[alpine sable]

You sure you are not missing one term?

[thorny root]

or starting at 10

[burnt escarp]

This is the question

[alpine sable]

Well

[thorny root]

,w solve pattern 0,3,7,15

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by biggboy#8536

[thorny root]

huh

[thorny root]

probably too complex to be the answer

[alpine sable]

https://oeis.org/search?q=0%2C3%2C7%2C15&sort=&language=&go=Search

I think it's a little too complicated for your level

It may be a mistake

[thorny root]

possibly one of these 3 if it were to be but can't see the relevancy of them to the question

I take back calling n(n-1)(n-2)/8 too complex

it's just irrelevant

I think

other two maybe a little complex

[lone heart]

@burnt escarp Has your question been resolved?

[lone heart]

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[grim pelican]

idk what this means sum1 help pls🙏🏻🙏🏻

[serene junco]

You need a number that's between them. Bigger than the smaller one, but smaller than the bigger one.

[grim pelican]

wdym

[serene junco]

Same thing as "Write a number between 1 and 10"

[grim pelican]

5

[serene junco]

Just with fractions

[grim pelican]

OHHH

damn ty

wait so is the first one

3/5

[serene junco]

That'll work.

There isn't just one right answer

But yes

[grim pelican]

then

is c

3/4

[serene junco]

sure

[grim pelican]

is

e

6/5

[serene junco]

No, that's greater than 1

[grim pelican]

can u help with question 8

[grim pelican]

what

[serene junco]

Might be easier if you write them as decimals

[grim pelican]

help me with no.8

[grim pelican]

then what would u write

damn

i js wrote 4/6 for e

someone help me w no.8 im very desperate

🙏🏻🙏🏻

😢

[serene junco]

#8 says you have six fractions

The first one is 2/8

and then each one after that is 1/8 more than the last one

So what would come next?

[grim pelican]

huh

uh

3/8

[serene junco]

yep

then just keep going

[grim pelican]

so whats the answer

[serene junco]

I'm not going to give you the answer. I'm positive you can figure it out.

Six fractions. They said the first fraction is 2/8, and asked you for the rest of them

[grim pelican]

oh

[serene junco]

You said 3/8,

just keep going

[grim pelican]

so u go all the way up to

6/8

[serene junco]

a bit further I think

because it started at 2/8

2/8, 3/8, ...

keep going until you have six fractions

[grim pelican]

bruh

oKay

[lone heart]

@grim pelican Has your question been resolved?

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[stuck geyser]

What does a double lined double pointed arrow mean?

[stuck geyser]

I'm thinking if and only if- but then my book goes onto literally separate the two and prove the => and the <= sides and it is confusing me

[pliant cedar]

if P is some statement and Q is another statement

if P implies Q, u wud write P => Q

if Q implies P u wud write Q => P

if both implications are true, u wud write P <=> Q, and say P if and only if Q

[stuck geyser]

what does Q <= P mean then?

[pliant cedar]

P => Q

same thing

the arrow poitns frmo P to Q

[stuck geyser]

Okay gotcha yeah

so my book is just doing it weird then

[pseudo ice]

Not quite

[pliant cedar]

well its not weird

[pseudo ice]

See here

[stuck geyser]

Ohh okay yeah

contextualizing it actually makes sense nw

[pliant cedar]

to say P implies Q, is the same thing as saying Q is implied by P

Q is implied by P can be written Q <= P

[stuck geyser]

Right

alrighty thanks for the wisdom

!close

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[pastel schooner]

Hey am I doing this right? I get decimals but calculator not allowed so I assume I am incorrect.

[pliant cedar]

what is csc(pi/6)

what is it equal to

and what is cot(pi/6)

[pastel schooner]

2?

[pliant cedar]

yeah

[pastel schooner]

I am not familiar with the radians for csc and cot tbh

[pliant cedar]

u know csc = 1/sin

and cot = 1/tan

[pastel schooner]

ah

fair

[pliant cedar]

what is cot(pi/6)

[pastel schooner]

sqrt(3)

[pliant cedar]

yeah so u first have ln(2+sqrt(3))

then the next one

[pastel schooner]

Ah so i did got it right just didnt put into radians right?

[pliant cedar]

? i dont get what u mean by "didnt put into radians"

[pliant cedar]

just use these

[pastel schooner]

1 sec

So i got these

I am not use to working with ln

Is a property the ability to just take the insides of each?

and use those

so ln ( (-2 - sqrt(3) ) + (2*sqrt(3) + sqrt(3) / 3) )

?

[pliant cedar]

$\ln\left(\frac{3\sqrt{3}}{3}\right)-\ln\left(2+\sqrt{3}\right)$ is what u have

[ocean seal]

SilverSoldier

[pastel schooner]

right

[pliant cedar]

$\ln\sqrt{3}-\ln\left(2+\sqrt{3}\right)$

[ocean seal]

SilverSoldier

[pliant cedar]

use laws of logarithms

[pastel schooner]

ah

you devide dont you

[pliant cedar]

yeah

[pastel schooner]

would you devide the first term by the second term

or second term divide first term?

[pliant cedar]

$\ln A-\ln B=\ln\frac{A}{B}$

[pastel schooner]

A/B right?

[ocean seal]

SilverSoldier

[pastel schooner]

Seems that it is the wrong answer

compared to the solution

[pliant cedar]

u can simplify ur answer

what answer do u get

[pastel schooner]

But I get sqrt(3) / 2 + 1

if i do a/b

[pliant cedar]

oh it looks like u must have ln(2+root3) - ln(root3) then

oh yeah

[pastel schooner]

i see my mess up

[pliant cedar]

$\int_a^b f(x),dx=F(b)-F(a)$, if $F$ is the indefinite integral of $f$

[ocean seal]

SilverSoldier

[pliant cedar]

uve done $F(a)-F(b)$

[ocean seal]

SilverSoldier

[pastel schooner]

Thank you for your help. I need to study up on my csc and cot

cause I suck at that

ty

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[flint pecan]

what have you tried?

[upper island]

I got the solution through long division method

[flint pecan]

and what is that method?

well my idea is to rationalize x first

[flint pecan]

then you'd get x=(sqrt3-1)/2

[flint pecan]

that seems good

[upper island]

I need a faster way

30-1 min solution

30 sec -1 min solution

[flint pecan]

the long division one shouldnt take that long

[upper island]

?

I tried to get the solution through value putting

But couldn't

[flint pecan]

or if you can spot right away that: 4x^3-2x^2-8x+5 = (4x^3+4x^2-2x)+(-6x^2-6x+3)+2

though this is the same logic as long division

[upper island]

Im sorry can you write it somewhere

I can't understand like that

[flint pecan]

$4x^3-2x^2-8x+5 = (4x^3+4x^2-2x)+(-6x^2-6x+3)+2$

[ocean seal]

jay.

[flint pecan]

im on a subway and i have no pen and paper with me

[upper island]

No that's good enough

Thanks

[raw jetty]

maybe you can factor it to make it easier

[flint pecan]

you can't factor it

[upper island]

Is it correct?

[flint pecan]

,rotate

[flint pecan]

what's the thing you wrote in front of (4x-1)

[upper island]

4x

[flint pecan]

uh then youre wrong

it should be

[upper island]

Wait

Im sending it

I corrected it

[flint pecan]

no

just but 4 in front of the second bracket

[upper island]

Ohhhh

[flint pecan]

$2x^2(2x-1)-4(2x-1)+1=(2x-1)(2x^2-4)+1$

[upper island]

I get it

[ocean seal]

jay.

[upper island]

Sorry for that

[flint pecan]

but i don't see what you're trying to do with that

[upper island]

I make rockie mistake too much

I was thinking about value putting

[flint pecan]

oh hm

[upper island]

Gottttt itt

Thank you so muchhhhhhh

[flint pecan]

ooh how?

can i see?

[flint pecan]

oh i see

[upper island]

Can we do it in shorter way?

More shorter way

[flint pecan]

hmm

i honestly prefer the long division

it only takes 3 lines to find out the remainder is 2

[upper island]

You are right

Its just that its been years since i have tried long division method

.close

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[tribal monolith]

what's the difference between displacement and distance

[vale wigeon]

displacement is how far your particle is from where it started, while distance is how far it travels

if you go 100m forward then 99m back your displacement will be 1m but your distance travelled will be 199m

[tribal monolith]

i see

.close

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[rocky grove]

Hmm

Multiply 4-3x by x^2 -x?

[raw jetty]

how does it help though? as its a remainder

heres a potentially clearer version

[rocky grove]

Hmm I'm thinking that maybe this can use polynomial remainder theorem

Not sure tho

[rich quiver]

@raw jetty see find the value of f(1) and f(-1) from the information given

[rich quiver]

Yeah it will work

[rocky grove]

Right?

Imma solve it myself just to make sure I get the idea

[raw jetty]

how do i do that?

this kinda just appeared from the questions out of nowhere

[rich quiver]

$f(x)=x(x-1)*g(x)+4-3x$

[ocean seal]

Anagh

[rich quiver]

By Division algo

Now put x=1

[raw jetty]

hold on

[rich quiver]

f(1)=0+4-3(1)

[raw jetty]

the book is leading me into something

[rich quiver]

Similarly find f(-1) using 2nd one

[raw jetty]

whats g(x)?

[rocky grove]

The quotient of f(x)/(x(x-1))

I'm not sure if I grasp the concept well

Hmm

Okay so you get f(1)

f(-1)

[raw jetty]

is finding x useful

x=4/3 and you can insert it into the g(x) function?

[rocky grove]

Finding what x?

[raw jetty]

the x

[rocky grove]

How did you find x=4/3?

[raw jetty]

because the x(x-1) from both the question and g(x) cancel out no?

the book gives these long formula

[rocky grove]

Hmm

[raw jetty]

the book shows these if $f(x)=(x-a)(x-b)$ then the remainder is $(x-a)×(h1(b)+f(a)$ where $h1(x)=\frac{f(x)}{x-a}$

[ocean seal]

Skill_Issue

[rocky grove]

Hmm

I think I found the answer

[raw jetty]

i think ill close this soon, this is basically like a book from my tutor and it feels out of my reach

[rocky grove]

No no wait

I found a good way

So

You know partial fractions?

[raw jetty]

like whst

[rocky grove]

Wait a min

[raw jetty]

so its like factoring but on fractions?

[rocky grove]

Yea

Wait

So

The question gave us the remainder of dividing x(x-1)

Which means that both x=0 and x=1 are not factors

Which also means that f(0) and f(1) are not 0

And so, we know that the remainder R(x) of the division will give us the value of f(1) at x=1 and f(0) at x=0

Also the question gave us another remainder

When dividing by x(x+1)

That means that it will give the value of f(x) at x=0 and x=-1

Then the question asks to find the remainder when we divide by (x-1)(x+1)

We already found f(1) and f(-1)

So we can say that $\frac{R}{(x-1)(x+1)} = \frac{A}{(x-1)} + \frac{B}{(x+1)}$

[ocean seal]

VulcanOne

[rocky grove]

While R equals f(1) and f(-1)

[raw jetty]

lemme try to process this

[raw jetty]

so your using that x(x-1) cannot be 0 as it has remainders, but how did you get f(0) or f(1) are not zero?

[rocky grove]

Because there was a remainder when we divided by them

[raw jetty]

oh

[rocky grove]

So like

$\frac{f(x)}{x(x-1)} = Q_1 (x) + \frac{4-3x}{x(x-1)}$

[ocean seal]

VulcanOne

[rocky grove]

Multiply both sides by x(x-1)

You get

[raw jetty]

fair

[rocky grove]

$f(x) = x(x-1) Q_1 (x) + 4-3x$

[ocean seal]

VulcanOne

[rocky grove]

When you plug in x=1 and x=0, the Q_1 (x) term doesn't matter any more

Since it will be multiplied by 0

Same thing for when dividing by x(x+1)

[raw jetty]

wait whwta R(x) again

[rocky grove]

Remainder function when we divide

[raw jetty]

o

[raw jetty]

how did you get this

[rocky grove]

Well we can decompose a polynomial that is being divided by another polynomial by partial fractions

[rich quiver]

Do you know division algorithm for poly?

[raw jetty]

no

[rich quiver]

Dividend = Divisor × Quotient + Remainder

@raw jetty do you know this

[raw jetty]

nope 😦

[rocky grove]

Hmm

[rich quiver]

Wtf

[rocky grove]

Can we say that 7=2*3 + 1?

[raw jetty]

although google for some reason shows me long dicision for polynomials

[raw jetty]

yep

[rocky grove]

Let's divide both sides by 2

We get 7/2 = 3+ 1/2

Right?

[raw jetty]

yep

[rocky grove]

Now this shows us that the quotient of dividing 7 by 2 gives 3

And the remainder of that division gives 1

Okay?

[raw jetty]

yeah, got it ^^

[rocky grove]

Now let's go to polynomials.

[raw jetty]

alright

[rocky grove]

Let's pick a really simple example

Can we say f(x) = x-2?

[raw jetty]

i think?

[rocky grove]

Just as an example

f(x) = x-2

[raw jetty]

yeah i guess

[rocky grove]

Now let's say we wanna divide it by x

Can you divide x-2 by x for me?

[raw jetty]

so f(x) divided by x is 1 with a remainder of 2?

[rocky grove]

-2 yeah

[raw jetty]

$1-\frac{2}{x}$

[ocean seal]

Skill_Issue

[rocky grove]

Nice okay

Now at x = 0, what does f(x) equal?

[raw jetty]

idk

[rocky grove]

The original f(x) = x-2

[raw jetty]

oh nvm

[rocky grove]

What does f(0) equal?

[raw jetty]

f(0)=1-2/0?

[rocky grove]

Wait no

[raw jetty]

huh

[rocky grove]

I want you to operate it on the original f(x)

[raw jetty]

this?

[rocky grove]

f(x) = x-2

This function

[raw jetty]

f(0)=0-2

=-2

[rocky grove]

Okay

Now I want you to notice that the remainder also equals -2

[raw jetty]

-2/0 :p

but yeah

[rocky grove]

No forget about -2/x

[raw jetty]

k

[rocky grove]

When we divided by x, the quotient was 1 and the remainder was -2

[raw jetty]

oh yeah yeah i forgot

[rocky grove]

And when we did f(0), we got -2

[raw jetty]

mhm

[rocky grove]

So we can say that the remainder equals the value of the function at the root of whatever we are dividing by (ie: x-r, r being the root, and f(r) = Remainder)

Now let's step it up a bit

Let's say we had $f(x) = x^2 -3x$

[ocean seal]

VulcanOne

[rocky grove]

And say we wanna divide it by x-5

Can you divide it for me?

[raw jetty]

uhh

is it like (x^2/x-5)-(3x/x-5)

[rocky grove]

Yeah but can you do it in long division?

[raw jetty]

ill try

x with a remainder of 2x?

is that correct or no

[rocky grove]

Hmm

Divide it more

[raw jetty]

x+2 with a remainder of -10

[rocky grove]

Yep

Now let's find f(5)

[raw jetty]

if we plug it here we get 10

[rocky grove]

Then the remainder we got was supposed to be +10

[raw jetty]

huh

let me try agaij

oh yeah

--10=+10

[rocky grove]

Yep

[raw jetty]

wait

so f(x) is like the remainder?

[rocky grove]

f(x) is the polynomial

When we divide it by some polynomial

We get a quotient and a remainder

[raw jetty]

a

[raw jetty]

btw why did you specifically ask for x-5 and x=5?

[rocky grove]

Just as an example to show that the remainder we get from dividing by x-5 gives f(5)

So that we can tackle the concept in the original problem

[rocky grove]

Which was this

[raw jetty]

wait

so if f(x) gets divided by a polynomial and say its x-3 then f(3)= the remainder?

[rocky grove]

Yep

Exactly

That's what Polynomial Remainder theorem says

[raw jetty]

ok

so how do we use this in the original problem

[rocky grove]

The question gives us the remainder

4-3x

And says that f(x) divided by x^2 -x gives a remainder of 4-3x

[raw jetty]

then what :p

[rocky grove]

Can you factor out x^2 -x?

[raw jetty]

x(x-1)

[rocky grove]

Okay

[raw jetty]

wait so we plug in x=0 and x=1?

[rocky grove]

Yep

Plug it into the remainder

And use that fact to infer that f(1) = remainder at x=1 and f(0) = remainder at x=0

[raw jetty]

f(0)=x(x-1)+4-3x?

wait

f(0)=0(0-1)+4-3x

[rocky grove]

Suppose that Q(x) is the quotient from the division

[raw jetty]

wait no

[raw jetty]

what

[rocky grove]

So x(x-1)Q(x) + 4-3x = f(x)

[raw jetty]

oh

[rocky grove]

x(x-1) is the divisor, Q(x) is the quotient, and 4-3x is the remainder

[raw jetty]

so
f(0)=4
f(1)=4-3(1)=1?

[rocky grove]

Yep

[raw jetty]

ah

do i do the same with the second thing

[rocky grove]

Yep

x^2 + x

And remainder of 4-x

[raw jetty]

f(x)=x(x+1)Q(x)+4-x
f(0)=4
f(-1)=5

hold on brb give me like 5 mins

[rocky grove]

Okay

:)

[raw jetty]

im back

@rocky grove

[rocky grove]

Ayo welcome back

You got f(-1) and f(0) and f(1) now

[raw jetty]

then what

[rocky grove]

The question asks you to find the remainder when we divide f(x) by x^2 -1

[raw jetty]

thats (x+1)(x-1) so i need to use f(-1) and f(1) somehow?

[rocky grove]

Yep yep

[raw jetty]

how though

[rocky grove]

Here is where this comes in handy

https://cdn.discordapp.com/attachments/490557019623915520/1069529355446910986/1000057838007226480.png

[raw jetty]

whats R again

[rocky grove]

R is our remainder

[raw jetty]

so were trying to get R?

[rocky grove]

Yep

R as a function

We will use the fact that R = f(x) at x=1 and x=-1

[rich quiver]

$f(x)=(x+1)(x-1)*Q(x)+ R(x)$ and $R(x) = ax+b$

[ocean seal]

Anagh

[rich quiver]

This is more easy to understand

[rocky grove]

Yep this too

Actually

Yeah go with Anagh's method

[rich quiver]

You can put x = 1 and x=-1 and find value of a and b

[rocky grove]

R = ax+b

[rich quiver]

I don't think he get what partial fractions are

[rocky grove]

Yep

[rocky grove]

Well if he did then that will save him time later

But you're right

[raw jetty]

so if f(1) that would be R(x) which we got earlier where f(1) was 1?

[rocky grove]

Yep

[rich quiver]

In integration?

[rocky grove]

Anywhere he encounters partial fractions

Anyways

[raw jetty]

wait i mean R(1)

and f(-1) is 5 so R(-1)=5?

[rocky grove]

Mhm

Now use those 2 facts

And get R(x) = ax+b

Straight line equation moment

[raw jetty]

so a+b is 1 b-a is 5

[rocky grove]

Mhm

[raw jetty]

b is 3 and a is -2?

[rocky grove]

Yep yep

Now what is the remainder?

[raw jetty]

on the R(x) or the whole f(x)

-2a+3

[rocky grove]

Remainder is R(x) itself

[raw jetty]

oh so 3?

wait

yeah 3 i think