🍁・general-2

[real imp]

cool

[abstract scaffold]

here's an example of a trial error combination i could make (obviously this one won't result in the answer, but it's a combination to try)

2x * x = 2x^2 (left side of cross)

-2 * 2 = -4 (right hand side of cross)

[real imp]

isn't trial and error just plain worse than doing a strict method?

isn't it just inneficient?

[hollow wedge]

depends

[abstract scaffold]

join 2x and 2, 2x(2) = 4x
join x and -2, x*(-2) = -2x

add them together. 4x+(-2x) = 2x (which is not 7x)

[hollow wedge]

this is usually a lot faster

especially for simple constants

[abstract scaffold]

my brain tends to find the trial and error in my head instantly

especially when it's 2x^2, 3x^2, or 5x^2

[real imp]

you have the math virus

[abstract scaffold]

if it's 4x^2, the left hand cross could be 2x/2x or 4x/x

but that still is a pretty easy case

technially. with "ur method". if you have ax^2 + bx + c, you have to trial and error find two constants x,y such that b = x+y and x*(y) = a(c)

so my argument there is that they are both technically trial and error

one can be less brain worky tho than the other

https://www.youtube.com/watch?v=5QyeZ7KwFKg here are some other methods

[real imp]

and how would you factorize t^3+1 for example?

[abstract scaffold]

use the sum of cubes formula

[real imp]

i don't know it i think

[abstract scaffold]

a^3 + b^3 = (a+b)(a^2-ab+b^2)

you can verify it by expanding

[real imp]

but

[abstract scaffold]

difference of two cubes
a^3 - b^3 = (a-b)(a^2+ab+b^2)

[real imp]

i dont understand how that'd work here

ohhh wait

1^3 = 1

[abstract scaffold]

t^3 + 1 = t^3 + 1^3

[real imp]

ok

coolio

so itd be (t+1)(t^2+t+1)

right?

[abstract scaffold]

t^2 - t + 1

minus on the right factor

[real imp]

common factor t?

or

but why -1?

i dont get it

[abstract scaffold]

-1?

[real imp]

i mean -t

[abstract scaffold]

it's the formula. it's weird. but it works.

[real imp]

no not the formula, the result you gave me

[abstract scaffold]

because you wrote +t

and t^3 + 1 = (t+1)(t^2-t+1)

[real imp]

what

where did i write that

im so confused

[real imp]

wheres the minus

[abstract scaffold]

you're meant to write minus that's why i was correcting u

[real imp]

difference of two cubes
a^3 - b^3 = (a-b)(a^2+ab+b^2)
@abstract scaffold but

no thats not the one

a^3 + b^3 = (a+b)(a^2-ab+b^2)
@abstract scaffold ohhh ok

[abstract scaffold]

t^3 + 1 is the sum of two cubes

[real imp]

okok

i mixed the two

[abstract scaffold]

yes, it's very easy to mix the two, since they are very similar to each other.

here's a proof

first line: cubic expansion
second line: isolate a^3+b^3 on one side, and that will be equal to something
third line: notice some group of terms can be factorised
fourth line: notice there's now a common factor of (a+b)
fifth line: "simplifying the right factor"
last line: simple algebra

and in similar fashion, we can prove the difference of two cubes

[sweet cape]

@abstract scaffold do you remember the dual simplex exercise?

Hi everyone

[hollow wedge]

hi yuan!

soy rojo

[sweet cape]

Hi martillo

Soy nativo de english

[fathom flame]

mi tu

[abstract scaffold]

@sweet cape yeah?

[sweet cape]

Anyway, i copied and pasted the exercise that you made on sheets, I’m waiting if the professor just tell me the grate, but that was so hard!

[abstract scaffold]

like like 6 variables is stupid

[sweet cape]

I didn’t copy that because this exercise didn’t need to give him

Yeah

Because i didn’t see decisionsmaking

[abstract scaffold]

it'd take me ages to do by hand.

[sweet cape]

Lol

[abstract scaffold]

bloody like 6x6 matrix

[sweet cape]

My teammates just told me, “why do you have some errors about results in those boxes”

Hahahaha

That was hard, the reason of that is all of things to do any exercise, i didn’t see that

[ashen mist]

hola

[gusty ermine]

Maricon

[chrome spear]

@summer wolf We can use english here if you want haha

si es demasiado español en #🐸・ꜱᴘ∕principiante

[summer wolf]

All this is shit

I can’t even understand anything

And I’ve been doing non stop Duolingo for 64 days

And still nothing makes sense

☹️

[chrome spear]

ahaha, take it easy :p this stuff takes time

[summer wolf]

yeah but

64 days

[chrome spear]

what grammar rules have you been learning recently in duolingo?

[summer wolf]

That was old picture

uh

[chrome spear]

Cuál reglas de gramática

[summer wolf]

That is right now

[chrome spear]

😐 I dunno what grammar stuff that includes, looks like general themes to me haha

[summer wolf]

uh

Well

[chrome spear]

like, you know all the present tense conjugations?

[summer wolf]

Yo bebo = I drink
Yo como = I eat
Tu bebes/Tu comes = you drink you eat
El Bebe/el come = he eats he drinks

That’s all so far

That’s like my only verbs I know

@chrome spear

[chrome spear]

ahhh

I wonder if there are lists of super common spanish verbs online 🤔 there have to be lol

[summer wolf]

idk

[chrome spear]

it looks like right now, just expanding vocab would help

[summer wolf]

My friend tried to teach me verb endings

It’s complicated though so I didn’t really get it

[chrome spear]

because like, you can conjugate everything wrong but if you know the vocab, people will still get what you're saying

[summer wolf]

Oh

Yeah like

[chrome spear]

well, mostly :p not for the complicated sentences of course (no para las oraciónes complicadas, por supesto)

[summer wolf]

Sometimes I make it sound like cave,an

Caveman

Like

I BUY CAR

I EAT APPLE

I DRINK WATER

I BUY APPLE AND EAT APPLE

like

Caveman talk

Also I heard it’s April fools

[chrome spear]

ah, but I suppose caveman talk can still transfer ideas ;)

Yo aprendo español - I learn spanish
Yo soy nuevo - I am new
👀

[summer wolf]

So I think I have the wrong role

[chrome spear]

yeah, just leave it for now lol

it'll change back tomorrow

I thiiiink=

[summer wolf]

Yeah but

Tmrw it will say Spanish native

And I, not

This is all so confusing and

[chrome spear]

you can change that pretty easily with bot commands, shouldn't be too bad

[summer wolf]

It’s discouraging sometimes like

[chrome spear]

;iamnot spanish native

[merry river]

Iklaendia#6756 You don't have that role

[summer wolf]

My friend is taking Italian

[chrome spear]

;iamnot english native

[merry river]

Iklaendia#6756 You no longer have the English Native role.

[summer wolf]

And he has like 200 days and alllllllllll the Duolingo stuff for Italian

And he knows better spanish than me

Even though he is learning Italian

[chrome spear]

oof, ok maybe don't do what I did

[summer wolf]

What did u do

[chrome spear]

I tried removing roles, it disabled chat for a moment haha

[summer wolf]

Oh

I suck at spanish

[chrome spear]

well if you click on your name, it still says english native

[summer wolf]

I just need to do more Duolingo

[chrome spear]

it's just green

[summer wolf]

Oh

[chrome spear]

so, tomorrow it'll all return to normal, probably

[summer wolf]

K

[chrome spear]

I dunno. What do you think is discouraging about it?

[gusty ermine]

alguien me llamo?
discord me dío un ping pero no sé en cuál canal, o sea que comente un error

[median trail]

@real imp

[hollow wedge]

Has revisado todos los canales?

tal vez @real imp te etiquetó

[vital ginkgo]

It's Ping Eternal

[hollow wedge]

Pingternal

[gusty ermine]

significa que alguien lo llamó a todos?

o algo diferente? 😅

[vital ginkgo]

It's because of April fools

The ping doesn't disappear

[gusty ermine]

[real imp]

i hacked the server

sorry guys

[vital ginkgo]

[graceful basin]

Oh it's April fool's...

[exotic kite]

🆙 | Stephiroth leveled up!

[graceful basin]

I forgot what say it is

Lol what's that

[vital ginkgo]

Sorry, have anime to watch

See you later

[gusty ermine]

no te molestaré ver anime :o
adíos y gracias 😄

[graceful basin]

I've had cake set aside to eat for hours and forgot it was here until now

[gusty ermine]

y ahora se va a desaparecer

[fallow dock]

alguien tenía cuenta en Taringa! desde el 2017 o antes y tiene una cuenta de Steam asociada al mismo correo?

tengo unos 30 intentos de acceso a una cuenta de Steam que no uso

no concretados por el 2FA

y a un amigo le pasó lo mismo

la única lista en común en la que estamos según https://haveibeenpwned.com/ es de Taringa!

[sacred harbor]

[abstract scaffold]

@gusty ermine https://www.youtube.com/watch?v=1aLingC__P0 this video can also help with the "absolute value of log with integration"

and that derivative of ln(x) is not quite 1/x

it's actually if f(x) = ln(x)
then f'(x) = 1/x {x is a real number | x > 0}
and if g(x) = ln(-x)
g'(x) = 1/x {x is a real number | x < 0}

so a trap question might be.

"what is the derivative of ln(x) at x = -1"

people may say. well, derivative is 1/x then thus at x=-1, derivative at the point is 1/(-1) = -1

However, that's wrong because ln(x) doesn't exist at x =-1, thus, the derivative at that point actually does not exist.

[gusty ermine]

the graph of ln(x)'s domain is all real numbers >0

is that why?

[abstract scaffold]

yes

red is original f(x) and blue is f'(x)

notice none exist on the left hand portion

[gusty ermine]

so the derivative of ln(x) does not exist when x=o or x= negative number

that's great to know

[abstract scaffold]

yes, because how can you draw a tangent at those points

the derivative at a certain point is the value of the gradient of the tangent of a function at the certain point

so, the domain of the function is the domain of the derivative

[fallow dock]

mmm hay dos lobos

a wolf pack

[still radish]

legs eleven

[real imp]

I need help

How do I factorize this?

[hollow wedge]

do you have to do a lot of manual factorization?

[real imp]

yeah

[shrewd pendant]

Wtf

[real imp]

I know this, don't know how to apply case 2 to this one

[shrewd pendant]

That’s so many problems haha

[real imp]

it's practice

[shrewd pendant]

I was never a big fan of practice

[real imp]

only way to learn

for me at least

im math dumb

practice is the fun part though

nobody likes having to memorize theory

[hollow wedge]

hahahaha @shrewd pendant story of my life

I like grocking theory, not memorizing it.

[shrewd pendant]

It’s a bit of a problem for me haha

[real imp]

grokking?

that's a new word for me

nice

[hollow wedge]

as soon as I understand something I have a false sense of security that I know it already.

[shrewd pendant]

I never study or practice for things

[shrewd pendant]

I figure it out as I go

[real imp]

are you in university?

berban

[hollow wedge]

yeah its grok apparently

[shrewd pendant]

No I graduated

[hollow wedge]

I was shooting from the hip, didn't know how to spell it lol

[real imp]

from high school or

university

[shrewd pendant]

With good grades

So I guess it worked for me

Yeah I’m 28 now

[real imp]

because in high school you don't really need to study

i didn't, at least

ah nice well, im not that smart

also maths is my weakest subject

ever

[shrewd pendant]

Well I don’t think it’s that I’m smart

I do work but not for studying

I prefer poring over the theories

[real imp]

guys i need help with that exerciseeee :(((

[shrewd pendant]

Lol sorry to derail the conversation

[real imp]

it ok

[hollow wedge]

ill see if I can solve it

I got sidetracked hehe

[real imp]

danke schön

[shrewd pendant]

Yeah I don’t know how to solve it 😦

[hollow wedge]

where does that picture come from, the guide?

[real imp]

yeppers

@gusty ermine can show you the solution, which I don't want to look at until I'm finished

bc she has the answers sheet

[hollow wedge]

no I mean, where does the guide come from?

[real imp]

my prof

[hollow wedge]

ah ok

hmm, I have the solution, but I don't understand how to use the method

well I know that the constant factor is 3, so the multiplication of the two constants in the parentheses have to be 3, so the only option is 1 and 3.

[real imp]

could you show me how you did it?

i'm not getting it

[abstract scaffold]

I did trial and error cross method

but I'll try to do it the more normal way

[real imp]

ah wait

i think i got it lol

yeah i did

(2x+1)(4x+3)

[abstract scaffold]

exactly

[real imp]

and now im stuck on number 9

[abstract scaffold]

got it

trial and error again lmao

[real imp]

yeah that's basically what im trying rn bc its too hard

[abstract scaffold]

here are all the combinations

[real imp]

oh god give me a sec to understand that lmao

whats the result

there

how do i find it

(3x+1)(2x-6)?

[median trail]

https://discordapp.com/channels/243838819743432704/383377678385938432/695265186893922344

This method is awesome

[real imp]

isn't it the same as this one

[median trail]

ever since my teacher taught it one class at my algebra class, I use it

[abstract scaffold]

yes, it's the same method

[real imp]

cool

[hollow wedge]

A core in all of these is to find the factors of the coefficient of x^2 and the constant.

[abstract scaffold]

@median trail the one i use?

[real imp]

well i would've sworn i had already tried that combination

seems like i missed it

thanks fcatorce

[median trail]

@abstract scaffold yea

you save up a lot of time

[real imp]

hold up

biut

but

but

it doesn't verify

as in

wait

[hollow wedge]

breathe

[real imp]

oh no nvm

im dumb

[hollow wedge]

respirá che

[real imp]

3*3 is not 6

[abstract scaffold]

hahhaa

[real imp]

:D

kms

[abstract scaffold]

omg little errors like that

[real imp]

yeah i hate this

[abstract scaffold]

boy they can kill

[shrewd pendant]

I’m pretty sure we spent like a week on factoring

I’m surprised you guys have to do it so much

[real imp]

im just slow

[shrewd pendant]

No it’s not that

[real imp]

this is a precalc class im taking at university bc im dumb

[shrewd pendant]

We just used the quadratic equation mostly

I have no idea how to solve these either and I’ve taken most of the math classes

[real imp]

you probably just forgot

[hollow wedge]

yeah i never liked these

I always guess

[shrewd pendant]

I don’t think they emphasized it in my curriculums

[hollow wedge]

never knew a method

isnt there something called completing the square?

[shrewd pendant]

Yes but that only works for certain ones iirc

[hollow wedge]

oh yeah, that is a different thing

[real imp]

whomst'ved

how tf do i do THAT

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

[shrewd pendant]

Lmao

[real imp]

we tried doing that one with panda yesterday but neither of us knew how to

[shrewd pendant]

You have to factor?

[real imp]

hi vers

[median trail]

yes

[real imp]

yes berban

[median trail]

(a + b) (a^2 - ab + b^2)

[real imp]

but its at the power of 3

isn't that for the power of 2

[abstract scaffold]

"the more safe way"

[median trail]

[real imp]

"the more safe way"
@abstract scaffold wait what

people do it like that?

[abstract scaffold]

it's less trial and error.

[shrewd pendant]

Always practice safe factoring

[real imp]

you have to equal the outer product with the center one?

[abstract scaffold]

yes, that's the method they taught at my school.

[real imp]

you have to equal the outer product with the center one?
@real imp ?

[abstract scaffold]

do a*c first

then find the factors of a*c

find which pair of factors when added match with b

assuming quadratic in form of ax^2+bx+c

[real imp]

im

uh

can i stick to the cross one

why is that one safer

[abstract scaffold]

because in the 6x^2 -5x - 6
there were a total of 16 cases to trial and error

but when doing the "safer one". there's only 5 cases

[real imp]

oh wow

i should deffo use that one then

[abstract scaffold]

so in this case a*c = -36

[real imp]

and 4*9 = 36

so 36 - 36 = 0

therefore thats the combo

[abstract scaffold]

ignoring the negative sign for a bit
1*36
2x13
3x18
4x9
6x6

then slap a negative sign somewhere since it's -36... u see the 4x9 combo works

since -9 + 4 = -5

[real imp]

yeah idk what i said

[abstract scaffold]

thus can split -5x into -9x + 4x

[real imp]

can you?

wait what

ohhhhh right

[abstract scaffold]

6x^2 + 4x - 9x - 6

[real imp]

wow thats big brain

[abstract scaffold]

group in pairs

(6x^2 + 4x) +(- 9x - 6)

fiind highest common factor in each pair

2x(3x+ 2) -3(3x+2)
then further factorise
(3x+2)(2x-3)

technically if you "include negative", then there's a total of 10 cases. since you can slap a negative on either 4 or 9... but only one works

but i like to say there's 5 cases... because it's not that hard to do addition and subtraction in one go

@real imp i told you about t^3+1 a bit before on another day

@real imp question 14 is also similar

to t^3 + 1

x^3 - 27

made up example:

x^3 - 8 = ?

a^3 - b^3 = (a-b)(a^2+ab+b^2)

x^3 - 8 = x^3 - 2^3 = (x-2)(x^2+2(1)x+2^2)

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

[real imp]

ohhh so i gotta raise it to the same power

[rain ermine]

are you learning how to factorize stuff ticko

[real imp]

yes epic

[rain ermine]

this all seems familiar and unfamiliar to me lol

[abstract scaffold]

general cubics are harder to factorise, if not impossible.

[real imp]

wdym by general cubics?

[abstract scaffold]

ax^3 + bx^2 + cx + d

[real imp]

ahh so like the full form

ok

[abstract scaffold]

well, there is a cubic formula... but it has like imaginary numbers. so it's never taught

[median trail]

👀

interesting

[abstract scaffold]

good luck remembering that

[real imp]

ew

[median trail]

Actually the fact that multiplying (1-i)(1+i) gives you ((1)^2+ (1)^2) if I'm not mistaken blew my mind

[abstract scaffold]

yes.

[real imp]

well i mean

i*i = 1

[abstract scaffold]

i*i = -1

[real imp]

yeah that

[hollow wedge]

hahaha

[real imp]

xd

[hollow wedge]

😂

[median trail]

yeah but when working with real numbers you can't get a sum of numbers that are both raised to the second power

[real imp]

shhhhjkdadshjkdas

i have terrible memory

[abstract scaffold]

unless you allow "non-integer numbers to be factorised in"

[hollow wedge]

wdym ed?

jajajaja

good lord tacko!

control yourself

[real imp]

i am so sorry

[hollow wedge]

hahahaha

[median trail]

like you can't get this when working with real numbers

x^2 + 1

because that's equal to (x-i)(x+i)

like factorise it

I mean

[hollow wedge]

ah ye

[median trail]

Oh I see

if one of them is negative, does it still work

I guess so

since they are both squared

[real imp]

is this already factorised

[median trail]

then why are we taught that we can not factorise that? don't tell me it's because engineering™

[abstract scaffold]

because "it's not a real number" it doesn't exsit, it's not real

@real imp short answer: no

i only needed to factorise with complex numbers when doing complex analysis

[median trail]

@real imp no, when something is factorised is because it's equal to factors that you can not reduce anymore and they are multiplying each toher

[abstract scaffold]

contour integration e.g.

[median trail]

contour integration sounds familiar to mw

is it related to green's theorem?

or something like that

[real imp]

@real imp no, when something is factorised is because it's equal to factors that you can not reduce more and they are multiplying each toher
@median trail so like

how do i get the factor of those two parts

[abstract scaffold]

@real imp recall the "special case" in ur picture

[real imp]

ok 1s

[abstract scaffold]

it's the difference of two squares

[real imp]

oh so just (4t+9s)(4t-9s)?

well that kinda makes sense

[abstract scaffold]

expand that out to see if you got it correct

[real imp]

wait no

it's negative

its -9

so should it be different

[hollow wedge]

9*9=???

[eager wagon]

[real imp]

81

im

confused

oh

uh

yeah

(2t + 3s) (2t - 3s)

whoops

but like then this formula is wrong

[abstract scaffold]

yep. all good now.

it's not wrong

[real imp]

it should say sqrt(a) and sqrt(b)

[abstract scaffold]

no.

[real imp]

oh

no

nvm

its

ok

squared yeah

im blind

[abstract scaffold]

4t^2 - 9s^2 = (2t)^2 - (3s)^2

[real imp]

yeah

wait

no

how are those equal

[abstract scaffold]

the minus is outside

[hollow wedge]

[abstract scaffold]

-(3s)^2 = -(3)^2(s)^2

-9s^2

(2t)^2 = 2^2(t)^2 = 4t^2

4t^2-9s^2

[real imp]

but what i dont get

is how (2t)^2 can equal 4t^2

[fathom flame]

(a+b)(a-b)=a^2-b^2 a=4t, b=9s (4t)^2-(9s)^2=16t^2-81s^2

[hollow wedge]

2*2 is 4

[real imp]

yeah

[fathom flame]

(2t)^2=2^2*t^2

[real imp]

4*4 is 16

how is that equal

[hollow wedge]

(ab)^2 = a^2*b^2

[real imp]

yeah

[vital ginkgo]

What does ^ mean

[abstract scaffold]

exponent

power

potencia

[real imp]

is there a difference between (a * b)^2 and a * b^2?

[abstract scaffold]

yes, huge difference

[fathom flame]

yes

[real imp]

i thought you could jsut mix them

with multiplication

like i get it has priority but

[abstract scaffold]

if inside brackets, the WHOLE thing needs to be squared

[real imp]

idk

[fathom flame]

you distribute the exponent power to both variables

[real imp]

ok so in

a*b^2

[fathom flame]

if inside brackets

[real imp]

only the b gets powered

[hollow wedge]

yeah

[real imp]

hm ok

[hollow wedge]

a*b^2 = (a)*(b^2)

[real imp]

ok ok got it

thanks

so in

x^3-27
the factorization would be
(x-3)(x^2-3x+9)

right?

cube difference

[median trail]

no

(x-3)(x^2+3x+9)

[real imp]

right

thanks

[abstract scaffold]

https://www.youtube.com/watch?v=LmbqB_pfcDU here is a 4hour 37min video of factorising 100 quadratics

[median trail]

blackpenredpen?

[real imp]

yeah

[median trail]

hahaha

[real imp]

ive seen that guy do integrals for

cant remember how much

(dunno how to do integrals)

oh i get it

he uses a black pen and a red pen

[fathom flame]

xd

[real imp]

oh wow this is a cool way to present it

[abstract scaffold]

contour integration sounds familiar to mw
@median trail sorry. got buried. greens theorem tiene alguna relacion

[real imp]

ok so

i got that x^3+2x^2+x = (x^2+x)(x+1)

just by looking at it which is cool and all

but i dont know how i got it

is it any different from grade 2 ones?

[abstract scaffold]

x^3 + 2x^2 + x has a common factor of x
x(x^2+2x+1), then now you have to factorise that quadradic
x(x+1)^2

[real imp]

oh

yeah i completely ignored that

thank uuu

[abstract scaffold]

with your answer.

x^2 + x can be factorised further

so x(x+1)(x+1) = x(x+1)^2

how u arrived at (x^2+x)(x+1), i have no idea.

[real imp]

oh so

its not factorized yet

how u arrived at (x^2+x)(x+1), i have no idea.
@abstract scaffold idk but it equals it

[abstract scaffold]

not completely factorised

(x^2+x)(x+1) is partly factorised

[real imp]

yeah

ok so

the complete factorization

would be

x(x+1)^2

?

[abstract scaffold]

yes

[real imp]

alrighty

[abstract scaffold]

I see how you arrived at x^2 + x and x+1

not the normal way to think about cross method, but ahhaha

[real imp]

xd

[gusty ermine]

https://youtu.be/d35Olnq1N5U

[gusty ermine]

How can I find a number value for x when the equation is 2x^2-8x

[still radish]

it needs to be equal to something

to solve it

[gusty ermine]

@median trail hey pretty lady, if you have time, can you help me?

[still radish]

I think the question is wrong

at a guess I would set it equal to zero

wait no

I'm retarded

[gusty ermine]

did you figure it out?

[still radish]

minimum is at derivative = 0

[median trail]

you have to check the sign before the point that makes the derivative zero and after

[still radish]

x = 2

• 8 is the minimum i guess

[median trail]

if the signs are + and then -

you have a minimum

wait

the other wait around

if the signs are - and then + you have a minimum

[shrewd pendant]

Yep

[gusty ermine]

Yeah i see

thank you kind people!

[still radish]

f(x) = 2x^2 - 8x
f'(x) = 4x - 8 = 0
x = 2
f(x = 2) = 8 - 16 = -8

it's been a while

[gusty ermine]

Awesome

Thank you so much ❤️

[real imp]

isnt this the kind of polynomial that 14 said its basically impossible to factorize?

[median trail]

x = 1 works

[real imp]

?

[median trail]

it's not impossible to factorize

I just told you one of the factors

[real imp]

w...what

what do i do with x=1

i dont get it

[still radish]

we call it algebraic divison

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

to work out what it is in the blank bracket we do something called algebraic division

[real imp]

should i use ruffini

[fathom flame]

tickoo

[median trail]

@real imp yes, absolutely

[real imp]

tickoo
@fathom flame ?

[fathom flame]

hi ticko

[real imp]

hi

[abstract scaffold]

@real imp ok. for cubics. in the form of

ax^3 + bx^2 + cx + d

test values of the form h/k

where h is factors of a
and k are factors of d

I was too tired to even see there were cubics 15 and onwards haha

so in the case of 16.

it's d = -2, a = 1

factors of d are {-2,-1,1,2}

factors of a are {1,-1}

thus, values to test are {-2,-1,1,2}

test x = -2

[real imp]

what even

what

[abstract scaffold]

(-2)^3 -4(-2)^2 + 5(-2) - 2 = -8-16-10-2 which is not zero

[real imp]

someone in the engineering server had me do the full form of the cubic thing

[abstract scaffold]

thus (x+2) can't be a factor

[real imp]

forgot the name

and i got to the result

but i can not do that on an exam

its just too long

this is what i did apparently

[mortal dagger]

nice nick

[abstract scaffold]

if it's polynomial division, then that just takes time to get used to.

[real imp]

yeah ok so now im trying to do it by polynomial division

https://www.purplemath.com/modules/synthdiv.htm

sent me this website but i haven't been able to understand it yet

like i know how to do ruffini

i just dont understand how to apply it here

what i do know is that there is an evident root

which is 1

[abstract scaffold]

ok. so when testing out all the points. you see x = 1 works

[real imp]

because a + b + c + d equal 0 or 1

[abstract scaffold]

thus (x-1) is a factor

[real imp]

so (x-1)(x+...)(x+...)

thats as far as i got

[abstract scaffold]

ok.

[real imp]

with this method

also i just now realise that it's (x-1) because x=1

nice

[abstract scaffold]

the rest of the possible rational factors {-2,-1,2} didn't work

yes, because we set P(x) = 0

[real imp]

yeah i just did it because i was told to before

[abstract scaffold]

so that if P(a) = 0, then (x-a)(x+....).... = 0

[real imp]

the rest of the possible rational factors {-2,-1,2} didn't work
@abstract scaffold how and when did we prove that

[abstract scaffold]

see the cubic is in the form ax^3 + bx^2 + cx + d

so d/a = -2/1

[real imp]

ok yes

[abstract scaffold]

so possible factors of d {-2,-1,1,2}
possible factors of a {-1,1}

d/a = {2,1,-1,-2,-2,-1,1,2}
getting rid of repeat cases d/a = {-2,-1,1,2}

[real imp]

a = 1 because its the only way that the first factor is 1, right?

like, the x from ax^3 is x1?

[abstract scaffold]

so thus (x+2) not a factor
(x+1) not a factor
(x-2) not a factor

[real imp]

sorry, ax^3 = x1

[abstract scaffold]

we figured out (x-1) is a factor just by setting the cubic equal 0 and tested x=1 as a possible point

[real imp]

or used the rule

for evident roots

right

[abstract scaffold]

@real imp the method will make more sense on question 19

[real imp]

ok ok cool

so we guess 1

and it worked

[abstract scaffold]

yep. and the others ones didn't so we NOW HAVE TO use polynomial division

[real imp]

why can't we keep going on guessing?

do we always stop at +-2?

or do we always stop at d?

[real imp]

LMFAOOO

[abstract scaffold]

we always stop when we exhausted the list of rational factors

[real imp]

but -2/3 for example is still rational

or or

or

uh

nkjgdjkjkdjkbnfkskdsldjslkjflsfjs

I DONT GET IT AAAAA

ljasdlkas

why did i have to be born so slow

ok i get that formula

what i dont get is

what limits x

to {2,1,-1,-2,-2,-1,1,2}

ohhhhhhhhh

so its

ok so

x = d

right?

its xd-1

and since its the denominator

it has to be diff from 0

wait

x not equals d

wahjkfhsdkfhsdkj

[abstract scaffold]

in this case. factors of d and factors of a

d = -2

ignore negative for a sec

[real imp]

ok d' = 2

[abstract scaffold]

factors of 2 are {-2,1,1,2}

[real imp]

yes

[abstract scaffold]

now factors of a
a = 1
{-1,1}

[real imp]

ok

[abstract scaffold]

take each term from {-2,-1,1,2} and divide it by the first term from {-1,1}

that is {-2/-1, -1/-1, 1/-1, 2/-1}

complete the list by taking each term and dividing by the second

[real imp]

-2/-1 = 2
-1/-1 = 1
1/-1 = -1
2/-2 = =-2
-2
-1
1
2

[abstract scaffold]

and get rid of redundant terms.... when it's a = 1.... we see this is fully redundant

[real imp]

Yeah

[abstract scaffold]

so u see in the end, list of x values to try are {-2,-1,1,2}

and when trying them, only x = 1 worked

only x=1 results the cubic to be zero

thus (x-1) is a zero

[real imp]

ok cool

by zero

you mean root

right?

[abstract scaffold]

yes. another name for root

zeroes, roots, factors

[real imp]

ohhhhh

so factors

are roots

well that would've been a nice thing to know two days ago when i started lol

[real imp]

thank you so much for telling me that like

i had no idea what a factor was

[abstract scaffold]

then u have to do this shit, where i totally didn't make any mistakes along the way.....

[real imp]

xd

is that a square root

[abstract scaffold]

nah, it's the dividing shitty thing

like we may have done in primary school

[real imp]

ah ok

[abstract scaffold]

@real imp oh fuck

im dumb

x = 2 works as well

[real imp]

yeah

[abstract scaffold]

ok, so polynomial division isn't needed

[real imp]

1 is actually a double root btw

[abstract scaffold]

i looked at my quadratic when division

and was like huh? i can factorise that

[real imp]

(x-2)(x-1)(x-1) are the factors

[abstract scaffold]

yep.

[real imp]

i just

am not sure about how i got there anymore

[abstract scaffold]

@real imp when u find two factors, the last one you can find

[real imp]

it was a looooong ride

[abstract scaffold]

so, x = 1, x = 2 worked

so (x-1)(x-2)(x-a) = 0

but (-1)(-2)(-a) = d

so -2a = 2

so a = -1

[real imp]

x(-1-2-a) = 0 right

[abstract scaffold]

if you expand out that cubic. you see those last three constant terms will be multiplied together to give "d"

[real imp]

wat

wait lemme

process

[abstract scaffold]

try expanding (x-1)(x-2)(x-a)

[real imp]

(x-1)(x-2)(x-a) =

= (x^2 - 3x + 2)(x-a) =
= (x^3 - 3x^2 + 2x -ax^2 + 3ax + 2a) =
= (x^3 - x^2(3 + a) + x(2+4a))

[abstract scaffold]

yep. try expanding it in the form of ax^3 + bx^2 + cx + d

and you'll hopefully see what I mean

you made a slight mistake on the second line

[real imp]

oh no

did i

[abstract scaffold]

(x-1)(x-2) expands to (x^2 - 3x + 2) {just like i found in my polynomial division}

[real imp]

ah no yeah i just

forgot to

add

well substract in this case

[abstract scaffold]

so, the -3 should be +2

[real imp]

oh that

yeah there i fucked up whoops

[abstract scaffold]

well, this is how it'd partially look like if done using division

but if 2 roots can be found, division can be avoided

[real imp]

ok so

(x-1)(x-2)(x-a) =

= (x^2 - 3x + 2)(x-a) =
= (x^3 - 3x^2 + 2x -ax^2 + 3ax + 2a) =
= (x^3 - x^2(3 + a) + x(2+4a))

right

i will go make dinner

[abstract scaffold]

don't forgeting the +2a at the end

[real imp]

i need a break

like rn

or im gonna fall asleep

[abstract scaffold]

cool, go do that

[real imp]

sorry

[abstract scaffold]

literally the only thing wrong now is the +2a term.
(x^2 - 3x + 2)(x-a) expands to

(x^3 - 3x^2 + 2x -ax^2 + 3ax - 2a)

a more general form can be found for the product of the roots {and the sum of the roots}

so another way to appraoch it is to use the product of roots formula or sum of roots formula

roots are x_1 =1 and x_2 =2
thus 1 + 2 + x_3 = -b/a = -(-4)/1 = 4
1 + 2 + x_3 = 4
=> 3 + x_3 = 4
x_3 = 1
so roots are x_1 = 1, x_2 = 2, x_3 = 1
thus factored form is:
(x-2)(x-1)^2

alternatively (product of roots)
x_1 = 1, x_2 = 2

(1)(2)*x_3 = -d/a = -(-2)/1 = 2
(1)(2)(x_3) = 2
2(x_3) = 2
x_3 = 1

thus roots are x_1 = 1, x_2 = 2, x_3 = 1
(x-1)^2(x-2)

[abstract scaffold]

@real imp the method we're using now for cubics works in general for quartics or even any polynomial of order n.

[wheat ravine]

are we allowed to share server links? there is a math server im apart of that specializes with helping people

[median trail]

@wheat ravine You can if you ask an admin first, and since I am one, then yeah go ahead

👀 I want to see

[wheat ravine]

https://discord.gg/DeE36D

i almost feel like this server in the math one go together

i really love it how with discord i can find and talk to people who know certain subjects crazy well

it feels amazing to be able to just find people who know what they're doing with things and get them to explain stuff

[median trail]

Ah I already joined that server

[junior thicket]

Hi can someone help me with algebraic fractions(?

[median trail]

I can tomorrow after 12 pm

[junior thicket]

Ok

[fathom flame]

u can join that server

[junior thicket]

Id like to distance myself from math as much as possible

[fathom flame]

algebraic fractions

[junior thicket]

Fracciones algebraicas

Ni idea si se dice así xd

[fathom flame]

what du need help with?

[abstract scaffold]

hello

[fathom flame]

Hi F14

[junior thicket]

Theres an exercise titled "fracciones algebraicas" and the first exercise is just
2x+5/5×+3
And idk what im supposed to do. Simplify them?? Divide??

The teacher isnt online so :(

[abstract scaffold]

it just says (2x+5)/(5x+3)?

like this?

[junior thicket]

Yea like that

And no other instruction

[abstract scaffold]

erm... i cant really think of anything if there's no instruction

I did polynomial division, but that it ends up not too pleasant

[junior thicket]

thanks for trying
Guess I'll just wait for him to get online tomorrow

[abstract scaffold]

other things i could possibly do to it is integrate or differentiate it. but there's no instruction, asi que no se que deberia hacer

[still radish]

Ozark está buena

pero no es pg

[eager wagon]

😛

[still radish]

@eager wagon qué significa la bufanda?

[eager wagon]

Es del autismo.. es el mes de concientización del autismo

[still radish]

bien. No lo sabía

[rain ermine]

What is broadband

[still radish]

it's a very wide elastic band

[eager wagon]

broadband = internet, Epic *

[real imp]

so uh

my gf did this without even blinking twice

she solved in 5 minutes what it took me 3 days what it took me to understand

What is broadband
@rain ermine the maximum internet speed capacity.
for example, if you have idk 1GB/s broadband
two people can use about 500MB /s
four people can use about 250MB/s

[rain ermine]

Oh

[abstract scaffold]

@real imp if you think when you find 1 root by hand and can do synthetic division just fine afterwards, stick with that.

I tend to make errors with polynomial division, so I try to exhuast my list first.

[real imp]

idk seems like she first used the constant

and then the evident root

which makes a lot of sense

since

(thing)+constant=0
(thing)=-constant

therefore 0=-constant

[abstract scaffold]

yeah. she used division each way through

[real imp]

thats nice

[abstract scaffold]

https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut37_syndiv.htm

synthetic division

synthetic division only works when dividing by (x-a)

so it seems

[abstract scaffold]

no, it works for all. it just mindfucking for me, because i'm not used to it.

[glass dragon]

No you're right

It works to check whether or not something is a zero for any factor

But to actually divide and get the dividend, it has to be a linear factor of the form x-a

[abstract scaffold]

i never learnt it in high school. wish i did. it's so much quicker.

plus it's technically less error prone since your adding rather than subtracting

i'll probably still stick to normal if it's not a monic

[glass dragon]

Yes but it's limited in application due to only be usable with a linear factor of the form x-a

[abstract scaffold]

mhmm, so i still have to have the traditional way in my toolbelt still

[vestal pine]

... course

[junior crystal]

#811515 📣 https://www.youtube.com/playlist?list=PLeA5t3dWTWvvwf5fw0Nl7mVk0OUjP1Ln2

[vestal pine]

@dire verge do you use this

[dire verge]

@vestal pine ya are even suppose to do with this

what are we *

[still radish]

I recommend listening to it but you could also try watching it with the sound off (results may vary)

[vestal pine]

[bold gull]

... Xd

[junior crystal]

#1068482 📣 @alpine cradle

[bold gull]

¿Qué?

[alpine cradle]

wtf

[bold gull]

... Xd

[junior crystal]

#1068482 📣 @alpine cradle

[alpine cradle]

@copper fox

[bold gull]

XD

[alpine cradle]

... rocco

[junior crystal]

#1079276 📣 xD

[bold gull]

... Ele

:C

[alpine cradle]

estás aquí?

Katakana

cómo se pronuncia tu nombre

Hoke o algo

[copper fox]

Pq el ping

[alpine cradle]

Puzzle

[eager wagon]

@copper fox

[alpine cradle]

.qdel 1079276 por favor haha

... staff

[junior crystal]

#1069850 📣 @copper fox

[copper fox]

What?

[alpine cradle]

... xd

[junior crystal]

#1068482 📣 @alpine cradle

[alpine cradle]

delet dis

lol i gabe you the wrong id

[copper fox]

.. xd bleh

[junior crystal]

Puzzle#3740 Quote Added

[copper fox]

... xd

[junior crystal]

#1079437 📣 bleh

[copper fox]

Eso?

[alpine cradle]

1068482

[copper fox]

I have no idea what you want me to do

And i dont think i have powers to delete things in this channel anyway

[alpine cradle]

you've created it so you can delete it haha

[copper fox]

I just changed it so it doesnt tag you anymore

.qdel 1079276

[junior crystal]

No quotes found which you can remove.

[alpine cradle]

the other one

1068482

there are 2 rn and the bot chooses one randomly haha

[copper fox]

... xd

[junior crystal]

#1068482 📣 @alpine cradle

[copper fox]

Oh

[alpine cradle]

xd

.qdel 1069850

[junior crystal]

Quote #1069850 deleted.

[copper fox]

. qdel 1068482

[alpine cradle]

sin espacio

[copper fox]

.qdel 1068482

[junior crystal]

Quote #1068482 deleted.

[alpine cradle]

te quiero hombre

[copper fox]

Lo siento

[alpine cradle]

está bien jaja

tenemos un polaco en #🐸・ꜱᴘ∕principiante

[copper fox]

🇵🇱

[still radish]

who does he follow orders from? @eager wagon

él es el jefe no?

[eager wagon]

Tarkin * ?

[still radish]

oh

i always assumed he was the boss

[real imp]

guise

ok so i factorized

and got to

[eager wagon]

https://www.youtube.com/watch?v=CICIOJqEb5c

RIP

[real imp]

(x-1)(x+4)(2x+1)

can I go further?

i tried solving the last equation

2x+1 = 0
2x = -1
x = -1/2

but when i calculate

(x-1)(x+4)(x+1/2)

which doesn't give me the right result

[vital ginkgo]

Notting Hill intesifies

[real imp]

the original polynome is

2x^3 + 7x^2 - 5x - 4

[fathom flame]

(x+1/2) or (2x+1)?

[real imp]

yeah exactly

i dont know if i can just leave it at

2x+1

or if i have to go further

if i leave it at 2x+1 the result is correct

[fathom flame]

@real imp you're missing the 2: (x-1)(x+4)2(x+1/2) if ur tryna to do that but its already factored in

[real imp]

what

no but wdym

i dont get it

[fathom flame]

i mean you're trying to factor the 2, right?

[real imp]

that is what i did

[fathom flame]

mm you can use ruffini

[real imp]

that's what i did

[real imp]

... thats what i did bruh

[fathom flame]

then use cross method for 2x^2+9x+4

(x-1)(2x+1)(x+4)

[real imp]

that's

what i did

i mean, i used ruffini but its the same

entendiste lo que pregunte?

[fathom flame]

estas hallando una de las raices (ceros)

al hacer: 2x+1=0

[median trail]

hi

what's going on 👀

oh ruffini

[real imp]

estas hallando una de las raices (ceros)
@fathom flame y bueno

Los factores son las raíces

[median trail]

not exactly, each factor helps you get the zeros

[fathom flame]

Los Factores son los números que se multiplican para obtener otro número

15=5*3

[median trail]

yes

5 and 3 are factors

[real imp]

ok but in (x + a), -a is the root

[median trail]

yes

[real imp]

Because

x+a = 0
x = -a

[median trail]

yes

what are you finding the roots of?

(x-1)(x+4)(2x+1)
this is okay

[real imp]

this is okay
@median trail that was my question, thank uuuu

[median trail]

you could factor out the 2 but it's not necessary

since the factor is linear

[gusty ermine]

@median trail if you have time, can you explain this to me? I don't even understand the question

[shrewd pendant]

🤔

Oh wait I get it

[median trail]

Oh that's not my area of expertise

sorry

[shrewd pendant]

This must be calculus then?